# Encyclopedia

Specific entries of a matrix are often referenced by using pairs of subscripts.
In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, such as
$\begin{bmatrix} 1 & 9 & 13 \ 20 & 55 & 4 \end{bmatrix}.$
An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 4. Entries are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied. These operations have many of the properties of ordinary arithmetic, except that matrix multiplication is not commutative, that is, AB and BA are not equal in general. .Matrices consisting of only one column or row define the components of vectors, while higher-dimensional (e.g., three-dimensional) arrays of numbers define the components of a generalization of a vector called a tensor.^ In one of the scenes in the movie, there is a dimension shown consisting only of blank white space, the "potential" area for matrix programming .
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ Everyone is hooked up to one and the same Matrix; there are not unique matrices generated for each individual.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

Matrices with entries in other fields or rings are also studied.
Matrices are a key tool in linear algebra. .One use of matrices is to represent linear transformations, which are higher-dimensional analogs of linear functions of the form f(x) = cx, where c is a constant; matrix multiplication corresponds to composition of linear transformations.^ Everyone is hooked up to one and the same Matrix; there are not unique matrices generated for each individual.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Since the Matrix is one large perception in and of itself, the keys represent perception into any and all things with the Keymaker being the very creator of access to perception.
• The Matrix Mythology and Characters Homepage 7 January 2010 23:54 UTC www.briandemilio.com [Source type: Original source]

^ Hadamard ( Matrix  M1, Matrix  M2)           Takes the Hadamard product of 2 matrices which is an entry by entry multiplication of the two matrices.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

Matrices can also keep track of the coefficients in a system of linear equations. .For a square matrix, the determinant and inverse matrix (when it exists) govern the behavior of solutions to the corresponding system of linear equations, and eigenvalues and eigenvectors provide insight into the geometry of the associated linear transformation.^ Overall, then, the system portrayed in The Matrix parallels Gnostic Christianity in numerous respects, especially the delineation of humanity's fundamental problem of existing in a dreamworld that simulates reality and the solution of waking up from illusion.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ But, once one realizes that the causality in the Matrix world is only virtual, since causality is not built into our perceptional system, one can violate the Matrix's simulated causal laws.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Matrix) to rescue Morpheus, and ends up transformed into a Warrior with Heart and a Savior of his people.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

Matrices find many applications. .Physics makes use of matrices in various domains, for example in geometrical optics and matrix mechanics; the latter led to studying in more detail matrices with an infinite number of rows and columns.^ What the sight of the rows of numbers is meant to do is to remind us that Neo no longer believes in the Matrix illusion but understands it is a program, but even so, he should continue to see it.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ For example, one might object that the Matrix Hypothesis implies that a computer simulation of physical processes exists, but (unlike the Metaphysical Hypothesis) it does not imply that the physical processes themselves exist.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

^ When Morpheus gives Neo a more detailed picture of what the Matrix is and how it came to be, Neo freaks out.
• The Matrix: Coding Counter Racism 7 January 2010 23:54 UTC academic.udayton.edu [Source type: Original source]

Graph theory uses matrices to keep track of distances between pairs of vertices in a graph. .Computer graphics uses matrices to project 3-dimensional space onto a 2-dimensional screen.^ First principle: any abstract computation that could be used to simulate physical space-time is such that it could turn out to underlie real physical processes.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

.Matrix calculus generalizes classical analytical notions such as derivatives of functions or exponentials to matrices.^ Everyone is hooked up to one and the same Matrix; there are not unique matrices generated for each individual.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

The latter is a recurring need in solving ordinary differential equations. Serialism and dodecaphonism are musical movements of the 20th century that use a square mathematical matrix to determine the pattern of music intervals.
.A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old but still an active area of research.^ Extendible Local Matrix Hypothesis : I am hooked up to a computer simulation of a local environment in a world, extended when necessary depending on subject's movements.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

^ If the computer is still feeding systematic sensory-motor impulses into Neo's brain when he is plugged into the Matrix world, then he will see the world the program is producing in his visual system.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ On one way of doing this, a computer simulates a small fixed environment in a world, and the subjects in the simulation encounter some sort of barrier when they try to leave that area.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

Matrix decomposition methods simplify computations, both theoretically and practically. For sparse matrices, specifically tailored algorithms can provide speedups; such matrices arise in the finite element method, for example.

## Definition

.A matrix is a rectangular arrangement of numbers.^ A more modern meaning of "matrix" is based in mathematics: a rectangular arrangement of symbols.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[1] For example,
$\mathbf{A} = \begin{bmatrix} 9 & 8 & 6 \ 1 & 2 & 7 \ 4 & 9 & 2 \ 6 & 0 & 5 \end{bmatrix}.$
An alternative notation uses large parentheses instead of box brackets:
$\mathbf{A} = \begin{pmatrix} 9 & 8 & 6 \ 1 & 2 & 7 \ 4 & 9 & 2 \ 6 & 0 & 5 \end{pmatrix}.$
.The horizontal and vertical lines in a matrix are called rows and columns, respectively.^ With Sati's line we have all the information we need to fully construct what has happened with respect to the Matrix and the war.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

The numbers in the matrix are called its entries or its elements. To specify a matrix's size, a matrix with m rows and n columns is called an m-by-n matrix or m × n matrix, while m and n are called its dimensions. The above is a 4-by-3 matrix.
.A matrix where one of the dimensions equals one is also called a vector, and may be interpreted as an element of real coordinate space.^ In The Matrix one can come to know reality.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ If so, then the Matrix Hypothesis may not yield reality.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

^ The inhabitants of a matrix may also be deceived in that reality is much bigger than they think.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

.An m × 1 matrix (one column and m rows) is called a column vector and a 1 × n matrix (one row and n columns) is called a row vector.^ Cypher is so called because of what he does (decode the Matrix) and what he is_a clever encrypter of his own character and motives (no one can decode him till it is too late).
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Since we'll be discussing different kinds of Matrix, we need a name for the one depicted in The Matrix; Agent Smith refers to a First Matrix, so let's call the one we see the Second Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

For example, the third row vector of the above matrix A is
$\begin{bmatrix} 4 & 9 & 2 \\ \end{bmatrix}.$
Most of this article focuses on real and complex matrices, i.e., matrices whose entries are real or complex numbers. More general types of entries are discussed below.

### Notation

Matrices are usually denoted using upper-case letters, while the corresponding lower-case letters, with two subscript indices, represent the entries. .In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (non-italic), to further distinguish matrices from other variables.^ While Merovingian uses many keys for his own selfish lifestyle, Neo uses one special key - uniquely made for him and his unique purpose - in his attempt to save the human race.
• The Matrix Mythology and Characters Homepage 7 January 2010 23:54 UTC www.briandemilio.com [Source type: Original source]

^ Like so many other symbols in this trilogy, we encounter another set of three.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

^ In particular, semantically neutral terms are not (at least when used without semantic deference): such terms plausibly include "philosopher", "friend", and many others.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

An alternative notation involves the use of a double-underline with the variable name, with or without boldface style, (e.g., $\underline{\underline{A}}$).
The entry that lies in the i-th row and the j-th column of a matrix is typically referred to as the i,j, (i,j), or (i,j)th entry of the matrix. For example, (2,3) entry of the above matrix A is 7. For example, the (i, j)th entry of a matrix A is most commonly written as ai,j. Alternative notations for that entry are A[i,j] or Ai,j.
.An asterisk is commonly used to refer to all of the rows or columns in a matrix.^ To use a Marxist analogy, the Power Plant is the workplace whilst the Matrix is all the false promises, hopes and Ideals used to keep the workers (ie.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ So, is that the child is (1) use his mind to change the Matrix ’spoon’ object’s property (in programming sense) (2) Neo is reference to the same ’spoon’ object (3) Neo saw it bend.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

For example, ai,∗ refers to the ith row of A, and a∗,j refers to the jth column of A. The set of all m-by-n matrices is denoted M(m, n).
A common shorthand is
A = [ai,j]i=1,...,m; j=1,...,n or more briefly A = [ai,j]m×n
to define an m × n matrix A. Usually the entries ai,j are defined separately for all integers 1 ≤ im and 1 ≤ jn. They can however sometimes be given by one formula; for example the 3-by-4 matrix
$\mathbf A = \begin{bmatrix} 0 & -1 & -2 & -3\ 1 & 0 & -1 & -2\ 2 & 1 & 0 & -1\ \end{bmatrix}$
can alternatively be specified by A = [ij]i=1,2,3; j=1,...,4.
.Some programming languages start the numbering of rows and columns at zero, in which case the entries of an m-by-n matrix are indexed by 0 ≤ im − 1 and 0 ≤ jn − 1.^ COMPLIMENTS I have always hated the fact that whenever I find some site to explain some of the doubts about Matrix, it starts telling me crap about the philosphy and all.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ Smith says, "Some believed that we lacked the programming language to describe your perfect world."
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

^ What the sight of the rows of numbers is meant to do is to remind us that Neo no longer believes in the Matrix illusion but understands it is a program, but even so, he should continue to see it.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[2] This article follows the more common convention in mathematical writing where enumeration starts from 1.

## Basic operations

.There are a number of operations that can be applied to modify matrices called matrix addition, scalar multiplication and transposition.^ There is a small group "independent" humans who live outside the Matrix in place called Zion.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ Everyone is hooked up to one and the same Matrix; there are not unique matrices generated for each individual.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ The Wachowski brothers indicate that the names were "all chosen carefully, and all of them have multiple meanings," and also note this applies to the numbers as well (Wachowski chat).
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

[3] These form the basic techniques to deal with matrices.
Operation Definition Example
Addition The sum A+B of two m-by-n matrices A and B is calculated entrywise:
(A + B)i,j = Ai,j + Bi,j, where 1 ≤ im and 1 ≤ jn.
$\begin{bmatrix} 1 & 3 & 1 \ 1 & 0 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 5 \ 7 & 5 & 0 \end{bmatrix} = \begin{bmatrix} 1+0 & 3+0 & 1+5 \ 1+7 & 0+5 & 0+0 \end{bmatrix} = \begin{bmatrix} 1 & 3 & 6 \ 8 & 5 & 0 \end{bmatrix}$
Scalar multiplication The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c:
(cA)i,j = c · Ai,j.
$2 \cdot \begin{bmatrix} 1 & 8 & -3 \ 4 & -2 & 5 \end{bmatrix} = \begin{bmatrix} 2 \cdot 1 & 2\cdot 8 & 2\cdot -3 \ 2\cdot 4 & 2\cdot -2 & 2\cdot 5 \end{bmatrix} = \begin{bmatrix} 2 & 16 & -6 \ 8 & -4 & 10 \end{bmatrix}$
Transpose The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:
(AT)i,j = Aj,i.
$\begin{bmatrix} 1 & 2 & 3 \ 0 & -6 & 0 \end{bmatrix}^T = \begin{bmatrix} 1 & 0 \ 2 & -6 \ 3 & 0 \end{bmatrix}$
Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, i.e. the matrix sum does not depend on the order of the summands: A + B = B + A.[4] The transpose is compatible with addition and scalar multiplication, as expressed by (cA)T = c(AT) and (A + B)T = AT + BT. Finally, (AT)T = A.
Row operations are ways to change matrices. .There are three types of row operations: row switching, that is interchanging two rows of a matrix, row multiplication, multiplying all entries of a row by a non-zero constant and finally row addition which means adding a multiple of a row to another row.^ There are MANY people asking LOTS of “what about” type questions for the matrix movies.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ People come in from all different types of places and their path of entry is always unique.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ I have been playing with this for hours and would LOVE it if you added another matrix with Drum Beats to go along with the tones!

.These row operations are used in a number of ways including solving linear equations and finding inverses.^ What you will find instead are essays that both elucidate the philosophical problems raised by the film and explore possible avenues for solving these problems.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

## Matrix multiplication, linear equations and linear transformations

Schematic depiction of the matrix product AB of two matrices A and B.
.Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix.^ The only aspect of your work I didn’t “like” was that you fell neccesarry that one should seperate the two interpretations of the Matrix Trilogy.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ Everyone is hooked up to one and the same Matrix; there are not unique matrices generated for each individual.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ What the sight of the rows of numbers is meant to do is to remind us that Neo no longer believes in the Matrix illusion but understands it is a program, but even so, he should continue to see it.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot-product of the corresponding row of A and the corresponding column of B:
$[\mathbf{AB}]_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + \cdots + A_{i,n}B_{n,j} = \sum_{r=1}^n A_{i,r}B_{r,j},$
where 1 ≤ im and 1 ≤ jp.[5] For example (the underlined entry 1 in the product is calculated as the product 1 · 1 + 0 · 1 + 2 · 0 = 1):
\begin{align} \begin{bmatrix} \underline{1} & \underline 0 & \underline 2 \ -1 & 3 & 1 \ \end{bmatrix} imes \begin{bmatrix} 3 & \underline 1 \ 2 & \underline 1 \ 1 & \underline 0 \ \end{bmatrix} &= \begin{bmatrix} 5 & \underline 1 \ 4 & 2 \ \end{bmatrix}. \end{align}
.Matrix multiplication satisfies the rules (AB)C = A(BC) (associativity), and (A+B)C = AC+BC as well as C(A+B) = CA+CB (left and right distributivity), whenever the size of the matrices is such that the various products are defined.^ One can also combine the various hypothesis above in various ways, yielding hypotheses such as a New Local Macroscopic Matrix Hypothesis.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

^ The residual issue concerns the various remaining skeptical hypotheses on the table, such as the Recent Matrix Hypothesis, the Local Matrix Hypothesis, and so on.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

^ Other such terms include "matrix" and "envatted", as defined in this article.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

[6] The product AB may be defined without BA being defined, namely if A and B are m-by-n and n-by-k matrices, respectively, and mk. Even if both products are defined, they need not be equal, i.e. generally one has
ABBA,
i.e., matrix multiplication is not commutative, in marked contrast to (rational, real, or complex) numbers whose product is independent of the order of the factors. An example of two matrices not commuting with each other is:
$\begin{bmatrix} 1 & 2\ 3 & 4\ \end{bmatrix} imes \begin{bmatrix} 0 & 1\ 0 & 0\ \end{bmatrix}= \begin{bmatrix} 0 & 1\ 0 & 3\ \end{bmatrix},$
whereas
$\begin{bmatrix} 0 & 1\ 0 & 0\ \end{bmatrix} imes \begin{bmatrix} 1 & 2\ 3 & 4\ \end{bmatrix}= \begin{bmatrix} 3 & 4\ 0 & 0\ \end{bmatrix} .$
.The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.^ I think the Matrix Hypothesis should be regarded as a metaphysical hypothesis with all three of these elements.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

^ NEO The beauty of the movie (The Matrix) for me is that just like in the matrix people believe that the world that they live in, their perceptions, and all of their beliefs, even their own identity is real.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ M, int N)           Return a matrix of dimension MxN populated with all elements equal to 1.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

$\mathbf{I}_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}.$
.It is called identity matrix because multiplication with it leaves a matrix unchanged: MIn = ImM = M for any m-by-n matrix M.^ My goal with this bit of speculation is to get into the progression of what I call the Four Ages of the Matrix.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

^ Cypher is so called because of what he does (decode the Matrix) and what he is_a clever encrypter of his own character and motives (no one can decode him till it is too late).
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker product.^ Our ordinary intuition is that there's something valuable about the real deal that is missing in a Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ So if we are in an extendible local matrix, external reality still exists, but there is not as much of it as we thought.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

^ As the Matrix raises the problem of our knowledge of the external world_might this all be just a dream?_Cypher raises the problem of other minds_can we know the content of someone else's mind?
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[7] .They arise in solving matrix equations such as the Sylvester equation.^ Those in such a solitary Matrix will think they are in the real deal.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

### Linear equations

A particular case of matrix multiplication is tightly linked to linear equations: if x designates a column vector (i.e. n×1-matrix) of n variables x1, x2, ..., xn, and A is an m-by-n matrix, then the matrix equation
Ax = b,
where b is some m×1-column vector, is equivalent to the system of linear equations
A1,1x1 + A1,2x2 + ... + A1,nxn = b1
...
Am,1x1 + Am,2x2 + ... + Am,nxn = bm .[8]
This way, matrices can be used to compactly write and deal with multiple linear equations, i.e. systems of linear equations.

### Linear transformations

Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. .A real m-by-n matrix A gives rise to a linear transformation RnRm mapping each vector x in Rn to the (matrix) product Ax, which is a vector in Rm.^ It is true that in the Matrix they would not really be giving each other flowers, or really holding hands.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

Conversely, each linear transformation f: RnRm arises from a unique m-by-n matrix A: explicitly, the (i, j)-entry of A is the ith coordinate of f(ej), where ej = (0,...,0,1,0,...,0) is the unit vector with 1 in the jth position and 0 elsewhere. The matrix A is said to represent the linear map f, and A is called the transformation matrix of f.
The following table shows a number of 2-by-2 matrices with the associated linear maps of R2. The blue original is mapped to the green grid and shapes, the origin (0,0) is marked with a black point.
 Vertical shear with m=1.25. Horizontal flip Squeeze mapping with r=3/2 Scaling by a factor of 3/2 Rotation by π/6 = 30° $\begin{bmatrix} 1 & 1.25 \ 0 & 1 \end{bmatrix}$ $\begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix}$ $\begin{bmatrix} 3/2 & 0 \ 0 & 2/3 \end{bmatrix}$ $\begin{bmatrix} 3/2 & 0 \ 0 & 3/2 \end{bmatrix}$ $\begin{bmatrix}\cos(\pi / 6) & -\sin(\pi / 6)\\ \sin(\pi / 6) & \cos(\pi / 6)\end{bmatrix}$
Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps[9]: if a k-by-m matrix B represents another linear map g : RmRk, then the composition gf is represented by BA since
(gf)(x) = g(f(x)) = g(Ax) = B(Ax) = (BA)x.
.The last equality follows from the above-mentioned associativity of matrix multiplication.^ The problem of other minds, like solipsism mentioned above, is equally a problem in or out of the Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Although I am talking here specifically about the "original" First Age of the Matrix (i.e., not the 4+1 Age I mentioned above), much of what follows will apply to the new Garden as well.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

.The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors.^ What the sight of the rows of numbers is meant to do is to remind us that Neo no longer believes in the Matrix illusion but understands it is a program, but even so, he should continue to see it.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[10] Equivalently it is the dimension of the image of the linear map represented by A.[11] .The rank-nullity theorem states that the dimension of the kernel of a matrix plus the rank equals the number of columns of the matrix.^ Matrix  first, Matrix  second, Matrix  third)           Form a new matrix which is a vector of 3 matrices which have the same number of columns.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

^ M, int N)           Return a matrix of dimension MxN populated with all elements equal to 1.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

^ PowerMatrix ( Matrix  X)           Given a data column vector X, this function computes a matrix, Mx, of the independent coordinate monomials based on the degree and dimension of this implicit polynomial.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

[12]

## Square matrices

.A square matrix is a matrix which has the same number of rows and columns.^ What the sight of the rows of numbers is meant to do is to remind us that Neo no longer believes in the Matrix illusion but understands it is a program, but even so, he should continue to see it.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied.^ Everyone is hooked up to one and the same Matrix; there are not unique matrices generated for each individual.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ M1M2t ( Matrix  M1, Matrix  M2)           Multiply two matrices after transposing the second and return a new matrix containing the result.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

^ M1tM2 ( Matrix  M1, Matrix  M2)           Multiply two matrices after transposing the first and return a new matrix containing the result.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

A square matrix A is called invertible or non-singular if there exists a matrix B such that
AB = In.[13]
This is equivalent to BA = In.[14] .Moreover, if B exists, it is unique and is called the inverse matrix of A, denoted A−1.^ It exists now, only as part of a neural-interactive simulation, that we call the Matrix.
• The Matrix Decoded | 100777.com 7 January 2010 23:54 UTC 100777.com [Source type: Original source]

^ The world, Morpheus explains, exists "now only as part of a neural interactive simulation that we call the Matrix."
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ It exists now only as part of a neural-interactive simulation that we call the Matrix.
• The Matrix Transcript | Dialogue from the Movie 7 January 2010 23:54 UTC www.ix625.com [Source type: Original source]
• The Matrix: Script original 7 January 2010 23:54 UTC www.thematrixfr.com [Source type: Original source]

The entries Ai,i form the main diagonal of a matrix. The trace, tr(A) of a square matrix A is the sum of its diagonal entries. .While, as mentioned above, matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factors: tr(AB) = tr(BA).^ Matrix  first, Matrix  second)           Add two matrices and return a new matrix containing the result.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

^ M1tM2 ( Matrix  M1, Matrix  M2)           Multiply two matrices after transposing the first and return a new matrix containing the result.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

^ Hadamard ( Matrix  M1, Matrix  M2)           Takes the Hadamard product of 2 matrices which is an entry by entry multiplication of the two matrices.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

[15]
.If all entries outside the main diagonal are zero, A is called a diagonal matrix.^ After all in the "real" world, outside of the Matrix, nothing would be happening of interest except to scientists.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ After all, so long as his experiences will be pleasant, how can his situation be worse than the inevitably crappy life he would lead outside of the Matrix?
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.If only all entries above (below) the main diagonal are zero, A is called a lower triangular matrix (upper triangular matrix, respectively).^ With Sati's line we have all the information we need to fully construct what has happened with respect to the Matrix and the war.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

^ But all this makes sense only on the supposition that the Matrix is a dream machine, an imagination manipulator, not just a purveyor of sensory hallucinations.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ The world, Morpheus explains, exists "now only as part of a neural interactive simulation that we call the Matrix."
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

For example, if n = 3, they look like
$\begin{bmatrix} d_{11} & 0 & 0 \ 0 & d_{22} & 0 \ 0 & 0 & d_{33} \ \end{bmatrix}$ (diagonal), $\begin{bmatrix} l_{11} & 0 & 0 \ l_{21} & l_{22} & 0 \ l_{31} & l_{32} & l_{33} \ \end{bmatrix}$ (lower) and $\begin{bmatrix} u_{11} & u_{12} & u_{13} \ 0 & u_{22} & u_{23} \ 0 & 0 & u_{33} \ \end{bmatrix}$ (upper triangular matrix).

### Determinant

A linear transformation on R2 given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one.
The determinant det(A) or |A| of a square matrix A is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area (in R2) or volume (in R3) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
The determinant of 2-by-2 matrices is given by
$\det \begin{pmatrix}a&b\\c&d\end{pmatrix} = ad-bc,$
the determinant of 3-by-3 matrices involves 6 terms (rule of Sarrus). The more lengthy Leibniz formula generalises these two formulae to all dimensions.[16]
The determinant of a product of square matrices equals the product of their determinants: det(AB) = det(A) · det(B).[17] Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. .Interchanging two rows or two columns affects the determinant by multiplying it by −1.[18] Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix.^ From the scifi POV, Neo’s stopping the sentinels isn’t him touching the source, its him using these “powers” outside of the Matrix for the first time.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ These two aspects – his sys-admin ability and wireless communications ability - provide that rationale for Neo’s ability to communicate/attack/destroy other machines and programs from the source both internally in the matrix and externally (wirelessly) in the real world.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ I kind of thought it would be cool to use the "Tristan" chord to show this pivot at the end of these three Matrix movies.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

Finally, the Laplace expansion expresses the determinant in terms of minors, i.e., determinants of smaller matrices.[19] .This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula.^ Even if you've seen the Matrix three times, you'll want to see it again after reading the chat transcripts, or even after reading what follows.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ I’ve seen them on youtube, I use this one alot, but I’d really like to have an actual tone matrix.

^ Even if you have not seen the Matrix movies, you will find this an effective spiritual and mind-healing book.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

Determinants can be used to solve linear systems using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.[20]

### Eigenvalues and eigenvectors

A number λ and a non-zero vector v satisfying
Av = λv
are called an eigenvalue and an eigenvector of A, respectively.[nb 1][21] The number λ is an eigenvalue of an n×n-matrix A if and only if A−λIn is not invertible, which is equivalent to
det(A−λI) = 0.[22]
The function pA(t) = det(AtI) is called the characteristic polynomial of A, its degree is n. Therefore pA(t) has at most n different roots, i.e., eigenvalues of the matrix.[23] .They may be complex even if the entries of A are real.^ NEO The beauty of the movie (The Matrix) for me is that just like in the matrix people believe that the world that they live in, their perceptions, and all of their beliefs, even their own identity is real.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ And since they jointly require (almost by definition) the presence of suffering, it can be said more or less truly that we "define [even the best] reality through misery and suffering."
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

According to the Cayley-Hamilton theorem, pA(A) = 0, that is to say, the characteristic polynomial applied to the matrix itself yields the zero matrix.

### Symmetry

A square matrix A that is equal to its transpose, i.e. A = AT, is a symmetric matrix; if it is equal to the negative of its transpose, i.e. A = −AT, then it is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A = A, where the star denotes the conjugate transpose of the matrix, i.e. the transpose of the complex conjugate of A.
.By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; i.e., every vector is expressible as a linear combination of eigenvectors.^ SymmetricEigenValuesandVectors ()           Compute the eigen values and eigen vectors of this matrix which is assumed to be a real symmetric matrix.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

.In both cases, all eigenvalues are real.^ In the latter three cases, one is deceived about the reality of an object, about whether there is an elephant there at all.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[24] This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see below.

### Definiteness

 Matrix A; definiteness; associated quadratic form QA(x,y); set of vectors (x,y) such that QA(x,y)=1 $\begin{bmatrix} 1/4 & 0\ 0 & 1\end{bmatrix}$ $\begin{bmatrix} 1/4 & 0\ 0 & -1/4\end{bmatrix}$ positive definite indefinite 1/4 x2 + y2 1/4 x2 − 1/4 y2 Ellipse Hyperbola
A symmetric n×n-matrix is called positive-definite (respectively negative-definite; indefinite), if for all nonzero vectors x ∈ Rn the associated quadratic form given by
Q(x) = xTAx
takes only positive values .(respectively only negative values; both some negative and some positive values).^ The only negative aspect is that before he is reinserted he may experience some inner moral human pangs of good or bad.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ The only way to make some sense of the name is to think of the god, not as the producer of dreams, but as the one who has power over dreams: both to give them and to take them away.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[25] .If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.^ It exists now, only as part of a neural-interactive simulation, that we call the Matrix.
• The Matrix Decoded | 100777.com 7 January 2010 23:54 UTC 100777.com [Source type: Original source]

^ The world, Morpheus explains, exists "now only as part of a neural interactive simulation that we call the Matrix."
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ It exists now only as part of a neural-interactive simulation that we call the Matrix.
• The Matrix Transcript | Dialogue from the Movie 7 January 2010 23:54 UTC www.ix625.com [Source type: Original source]
• The Matrix: Script original 7 January 2010 23:54 UTC www.thematrixfr.com [Source type: Original source]

.A symmetric matrix is positive-definite if and only if all its eigenvalues are positive.^ But all this makes sense only on the supposition that the Matrix is a dream machine, an imagination manipulator, not just a purveyor of sensory hallucinations.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[26] The table at the right shows two possibilities for 2-by-2 matrices.
Allowing as input two different vectors instead yields the bilinear form associated to A:
BA (x, y) = xTAy.[27]

## Computational aspects

.In addition to theoretical knowledge of properties of matrices and their relation to other fields, it is important for practical purposes to perform matrix calculations effectively and precisely.^ I love the tone matrix, but what happened to the thin that had distortion pedals, tone matrices and other instrumental stuff?

^ Are they themselves effectively a node on the Matrix, sharing common brain elements with others?
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Constrained by his purpose in the Matrix of the Second Age, he seeks power related to controlling the traffic of data (i.e., "a trafficker of information").
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

The domain studying these matters is called numerical linear algebra.[28] As with other numerical situations, two main aspects are the complexity of algorithms and their numerical stability. .Many problems can be solved by both direct algorithms or iterative approaches.^ What you will find instead are essays that both elucidate the philosophical problems raised by the film and explore possible avenues for solving these problems.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

For example, finding eigenvectors can be done by finding a sequence of vectors xn converging to an eigenvector when n tends to infinity.[29]
Determining the complexity of an algorithm means finding upper bounds or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, e.g. multiplication of matrices. .For example, calculating the matrix product of two n-by-n matrix using the definition given above needs n3 multiplications, since for any of the n2 entries of the product, n multiplications are necessary.^ This makes perfect sense, given that his environment is the product of dreaming, since dreams consist of images and images are subject to the will.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Since we'll be discussing different kinds of Matrix, we need a name for the one depicted in The Matrix; Agent Smith refers to a First Matrix, so let's call the one we see the Second Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Hadamard ( Matrix  M1, Matrix  M2)           Takes the Hadamard product of 2 matrices which is an entry by entry multiplication of the two matrices.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

The Strassen algorithm outperforms this "naive" algorithm; it needs only n2.807 multiplications.[30] A refined approach also incorporates specific features of the computing devices.
.In many practical situations additional information about the matrices involved is known.^ Many have done this in one movie - I’m not sure I know of any others that have done it over three (there might be, but I doubt they are in genre flicks or I would have known about them).
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

An important case are sparse matrices, i.e. matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient method.[31]
.An algorithm is, roughly speaking, numerical stable, if little deviations (such as rounding errors) do not lead to big deviations in the result.^ And such error might lead to dramatic consequences: everyone around the person is really out to get him_his wife, friends, and so on.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

For example, calculating the inverse of a matrix via Laplace's formula (Adj (A) denotes the adjugate matrix of A)
may lead to significant rounding errors if the determinant of the matrix is very small. .The norm of a matrix can be used to capture the conditioning of linear algebraic problems, such as computing a matrix' inverse.^ Compute the inverse of this matrix and return the result as a new matrix.
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

^ MonomialVector ( Matrix  X)           Computes the monomial vector for a single data point using the matrix obtained from ImplicitPolynomial.computePowerMatrix(Matrix X) .
• Uses of Class math.Matrix (ShaRP - API Specification) 25 September 2009 2:36 UTC www.ece.uncc.edu [Source type: Reference]

[32]
.Although most computer languages are not designed with commands or libraries for matrices, as early as the 1970s, some engineering desktop computers such as the HP 9830 had ROM cartridges to add BASIC commands for matrices.^ It has to be some sort of control of the Matricians' intellectual powers, which we learn early on in the movie are free from the control of direct sensory-motor computer input.19 It must, then, be some sort of mind control.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices.^ If God is responsible, we would need to plead with him successfully, or to fight him and win; if it's the mathematical formulae (computer programs?
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ It wasn’t some computer program that woke up inside him!
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ Smith says, "Some believed that we lacked the programming language to describe your perfect world."
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

[33]

### Matrix decomposition methods

.There are several methods to render matrices into a more easily accessible form.^ In the real world to which Neo "awakes" and into which he will, we suppose, eventually lead everyone, there will be no more flying.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ There is easily room for a trilogy here, and more, as the ending is one of those 'it is only the beginning' kind of endings.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ The Creed doesn't really say any more than this, but it gets heavy interpretation in the Catholic Church [3] , so there are several variations on the whole story.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

They are generally referred to as matrix transformation or matrix decomposition techniques. .The interest of all these decomposition techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.^ In looking at the Matrix trilogy, I’d be interested in to hear how you see this similar analogy applying.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ This is # mightily similar to the process which is carried out here, # in the NDS. It is common for one to be puzzled as to the # stated assumptions of others, that one is are not where they # are, that one will awaken, to their true nature, to freedom.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ He didn't think about these philosophical questions at all.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

The LU decomposition factors matrices as a product of lower (L) and an upper triangular matrices (U).[34] .Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution.^ The pause draws on and the second hand of the clock ticks forward, and we descend once more, falling out of the presence of the Divine and back into the domain of time.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form.[35] .Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row.^ Both of them lock on, holding on to one another.

^ Everyone is hooked up to one and the same Matrix; there are not unique matrices generated for each individual.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ The films presentation of the matrix as a corporate network of human conceptions (or samsara) which are translated into software codes that reinforce one another illustrates this close relationship.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

Singular value decomposition expresses any matrix A as a product UDV, where U and V are unitary matrices and D is a diagonal matrix.
A matrix in Jordan normal form. The grey blocks are called Jordan blocks.
The eigendecomposition or diagonalization expresses A as a product VDV−1, where D is a diagonal matrix and V is a suitable invertible matrix.[36] If A can be written in this form, it is called diagonalizable. .More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into Jordan normal form, that is to say matrices whose only nonzero entries are the eigenvalues λ1 to λn of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right.^ "The main purpose of the Council on Foreign Relations is promoting the disarmament of US. sovereignty and national independence and submergence into an all powerful, one world government".
• The Matrix Decoded | 100777.com 7 January 2010 23:54 UTC 100777.com [Source type: Original source]

^ The only technical problem is how one would go about feeding a storyline directly into a brain.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ If one takes the metaphysical view, Neo becomes self-realized, and this has nothing to do with sentient programs or all the rest - but then the “belief” in his powers has to do with more metaphysical/religious connotations.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

[37] Given the eigendecomposition, the nth power of A (i.e. n-fold iterated matrix multiplication) can be calculated via
An = (VDV−1)n = VDV−1VDV−1...VDV−1 = VDnV−1
and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead. This can be used to compute the matrix exponential eA, a need frequently arising in solving linear differential equations, matrix logarithms and square roots of matrices.[38] To avoid numerically ill-conditioned situations, further algorithms such as the Schur decomposition can be employed.[39]

## Abstract algebraic aspects and generalizations

Matrices can be generalized in different ways. .Abstract algebra uses matrices with entries in more general fields or even rings, while linear algebra codifies properties of matrices in the notion of linear maps.^ We know that the machines used humans as a power source even though they had the technology to produce more effecient power sources (like “a form of fusion” mentioned in M1).
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

It is possible to consider matrices with infinitely many columns and rows. Another extension are tensors, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or two-dimensional array of numbers.[40] Matrices, subject to certain requirements tend to form groups known as matrix groups.

### Matrices with more general entries

This article focuses on matrices whose entries are real or complex numbers. However, matrices can be considered with much more general types of entries than real or complex numbers. As a first step of generalization, any field, i.e. a set where addition, subtraction, multiplication and division operations are defined and well-behaved, may be used instead of R or C, for example rational numbers or finite fields. For example, coding theory makes use of matrices over finite fields. .Wherever eigenvalues are considered, as these are roots of a polynomial they may exist only in a larger field than that of the coefficients of the matrix; for instance they may be complex in case of a matrix with real entries.^ The graph above only shows the relationship between the seven instances of Neo and the greater ages of the Matrix.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

^ I think that, in the same way, they’re saying that people in the real world don’t have free will, only the illusion of it.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ Consider the example of someone who has lived their entire life in the Matrix: when they talk of "chickens," they don't actually refer to real chickens; at best they refer to the computer representations of chickens that have been sent to their brain.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.The possibility to reinterpret the entries of a matrix as elements of a larger field (e.g., to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues.^ But what happens when the Matrix's version of reality is dissolved?
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ With Sati's line we have all the information we need to fully construct what has happened with respect to the Matrix and the war.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

^ So in choosing to return from the "real world" to the Matrix world, Cypher is just choosing between two systematic sets of appearances.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.Alternatively one can consider only matrices with entries in an algebraically closed field, such as C, from the outset.^ Your analysis is the only one which considers it from a scientific point of view.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

.More generally, abstract algebra makes great use of matrices with entries in a ring R.^ We know that the machines used to serve and protect humans and spared great expense to make sure they didn’t suffer in the first matrix.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

[41] .Rings are a more general notion than fields in that no division operation exists.^ The human body generates more bio-electricity than a 120-volt battery and over 25,000 BTU's of body heat.
• The Matrix Transcript | Dialogue from the Movie 7 January 2010 23:54 UTC www.ix625.com [Source type: Original source]
• The Matrix: Script original 7 January 2010 23:54 UTC www.thematrixfr.com [Source type: Original source]

^ THIS film has tangible cause and effects, and hokey emotional schmaltz, no more cheesy than a thumbs up from a T-800 on his way to oblivion.
• MATRIX REVOLUTIONS -- Ain't It Cool News: The best in movie, TV, DVD, and comic book news. 26 January 2010 0:55 UTC www.aintitcool.com [Source type: FILTERED WITH BAYES]

^ The human body generates more bio-electricity than a 120-volt battery and over 25,000 BTUs of body-heat.
• The Matrix Decoded | 100777.com 7 January 2010 23:54 UTC 100777.com [Source type: Original source]

The very same addition and multiplication operations of matrices extend to this setting, too. .The set M(n, R) of all square n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module Rn.^ This duality between the matrix and the reality beyond it sets up the ultimate goal of the rebels, which is to free all minds from the matrix and allow humans to live out their lives in the real world beyond.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ This duality between the Matrix and the reality beyond it sets up the ultimate goal of the rebels, which is to free all minds from the Matrix and allow humans to live out their lives in the real world beyond.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[42] If the ring R is commutative, i.e., its multiplication is commutative, then M(n, R) is a unitary noncommutative (unless n = 1) associative algebra over R. .The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible.^ Effectively Neo IS fighting every Smith in the Matrix, even though only one of them is actually engaged in direct combat.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ Neither the "stillness" of the pleroma nor the unchanging "nothingness" of nirvana are characterized by the dependence on technology and the use of force which so characterizes both of the worlds of the rebels in The Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ A brain port, along the lines of that in the Matrix, is not only a scientific best guess for the future; I am working on such a port now, and it will be with us within a decade at most.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[43] Matrices over superrings are called supermatrices.[44]
.Matrices do not always have all their entries in the same ring - or even in any ring at all.^ People come in from all different types of places and their path of entry is always unique.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

.One special but common case is block matrices, which may be considered as matrices whose entries themselves are matrices.^ Consider the scene in which Neo, the star ("The One") is finally surrendering to the persuasions of his new friends/rescuers, is sitting in a special chair.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ This makes some sense in the case of, say, a married man tempted to adultery, whose guilt may prevent him from full enjoyment.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

The entries need not be quadratic matrices, and thus need not be members of any ordinary ring; but their sizes must fulfil certain compatibility conditions.

### Relationship to linear maps

Linear maps RnRm are equivalent to m-by-n matrices, as described above. More generally, any linear map f: VW between finite-dimensional vector spaces can be described by a matrix A = (aij), after choosing bases v1, ..., vn of V, and w1, ..., wm of W (so n is the dimension of V and m is the dimension of W), which is such that
$f(\mathbf{v}_j) = \sum_{i=1}^m a_{i,j} \mathbf{w}_i\qquad\mbox{for }j=1,\ldots,n.$
.In other words, column j of A expresses the image of vj in terms of the basis vectors wi of W; thus this relation uniquely determines the entries of the matrix A.^ In his re-entry into the matrix, however, Neo retains the "residual self-image" and the "mental projection of [a] digital self."
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ On a sci-fi note, I’m wondering if you have a clear idea as to what actually happens when the matrix “reloads.” In other words, if I were in the Matrix, and this thing reloads, what would I notice?
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ In his re-entry into the Matrix, however, Neo retains the "residual self-image" and the "mental projection of [a] digital self."
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

Note that the matrix depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices.[45] .Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix AT describes the transpose of the linear map given by A, with respect to the dual bases.^ Many contemporary philosophers have discussed a similar skeptical dilemma that is a bit closer to the scenario described in The Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[46]

### Matrix groups

A group is a mathematical structure consisting of a set of objects together with a binary operation, i.e. an operation combining any two objects to a third, subject to certain requirements.[47] A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group.[nb .2][48] Since in a group every element has to be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups.^ More to the point of our general discussion: for all you know, you may well be trapped in the Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ In his training he learns that the Matrix, as a computer-generated group dream, can be manipulated by a human being.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ There is a small group "independent" humans who live outside the Matrix in place called Zion.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

.Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups.^ So, is that the child is (1) use his mind to change the Matrix ’spoon’ object’s property (in programming sense) (2) Neo is reference to the same ’spoon’ object (3) Neo saw it bend.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

For example, matrices with a given size and with a determinant of 1 form a subgroup of (i.e. a smaller group contained in) their general linear group, called a special linear group.[49] Orthogonal matrices, determined by the condition
MTM = I,
form the orthogonal group.[50] They are called orthogonal since the associated linear transformations of Rn preserve angles in the sense that the scalar product of two vectors is unchanged after applying M to them:
(Mv) · (Mw) = v · w.[51]
.Every finite group is isomorphic to a matrix group, as one can see by considering the regular representation of the symmetric group.^ Effectively Neo IS fighting every Smith in the Matrix, even though only one of them is actually engaged in direct combat.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ The only body he sees and moves is the one he has in the Matrix world.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ But by the end of the movie, Neo as the One (or the anti-one as Heidegger would see it), has only promised to give people in the Matrix freedom to bend the rules.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[52] .General groups can be studied using matrix groups, which are comparatively well-understood, by means of representation theory.^ By “using his powers outside the matrix” I mean that he was able to “exist” outside the matrix.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ More to the point of our general discussion: for all you know, you may well be trapped in the Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ In his training he learns that the Matrix, as a computer-generated group dream, can be manipulated by a human being.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[53]

### Infinite matrices

.It is also possible to consider matrices with infinitely many rows and/or columns[54] even if, being infinite objects, one cannot write down such matrices explicitly.^ If even one "average" person walks out of this movie with a slight change of perspective, it is of infinite value.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ One cannot help wondering if this dictum only holds within the matrix or if there is in fact "no spoon" even in the real world beyond it.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ Even our own culture has experienced many different worlds created by new interpretations of people and of nature that changed what counted as human beings and things.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry (these index sets need not even be subsets of the natural numbers).^ What the sight of the rows of numbers is meant to do is to remind us that Neo no longer believes in the Matrix illusion but understands it is a program, but even so, he should continue to see it.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ It is possible to suppose that all five of these people were genetically designed as well.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

^ Frequently in these films there are lines of dialogue that seem to carry a particular, superficial meaning but in fact are deep wells of symbolism.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

.The basic operations of addition, subtraction, scalar multiplication and transposition can still be defined without problem; however matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general.^ The rebel humans want to get to Zion (meaning "sanctuary" or "refuge"), but isn't the Matrix already a type of Zion_yet without the dubious virtue of generating true beliefs?
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ In his re-entry into the matrix, however, Neo retains the "residual self-image" and the "mental projection of [a] digital self."
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ In his re-entry into the Matrix, however, Neo retains the "residual self-image" and the "mental projection of [a] digital self."
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason.^ This may be the aim in a limited number of cases, but the goal for most AI developers is to make use of the ways in which robots can outperform humans_rather than those in which they can only potentally become our match.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.For a matrix A to describe a linear map f: VW, bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combination of basis vectors, so that written as a (column) vector v of coefficients, only finitely many entries vi are nonzero.^ I can only repeat myself so many times that the purpose of this essay is NOT to validate the scientific logic behind the matrix.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ Many contemporary philosophers have discussed a similar skeptical dilemma that is a bit closer to the scenario described in The Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

Now the columns of A describe the images by f of individual basis vectors of V in the basis of W, which is only meaningful if these columns have only finitely many nonzero entries. .There is no restriction on the rows of A however: in the product A·v there are only finitely many nonzero coefficients of v involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined.^ And no one, not even you, not .

^ Right now, there's only one rule : Our .

^ Right now there's only one rule, our way, or the highway.
• The Matrix Decoded | 100777.com 7 January 2010 23:54 UTC 100777.com [Source type: Original source]

.Moreover this amounts to forming a linear combination of the columns of A that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries, because each of those columns do.^ Many of our desires are derived from other desires plus belief, for instance if Ralph desires to kiss Grandma only because he desires an inheritance and he believes kissing Grandma is necessary to achieve this.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

One also sees that products of two matrices of the given type is well defined (provided as usual that the column-index and row-index sets match), is again of the same type, and corresponds to the composition of linear maps.
Infinite matrices can also be used to describe operators on Hilbert spaces, where convergence and continuity questions arise, which again results in certain constraints that have to be imposed. .However, the explicit point of view of matrices tends to obfuscate the matter,[nb 3] and the abstract and more powerful tools of functional analysis can be used instead.^ If one takes the metaphysical view, Neo becomes self-realized, and this has nothing to do with sentient programs or all the rest - but then the “belief” in his powers has to do with more metaphysical/religious connotations.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ This point however, as many of these points, is not explicit in the movie, exactly.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ It is the illusionary world used to control the human’s by tricking them into obedience while the Machines suck them dry; the “Power Plant” is a seperate setup with a seperate function (the actual sucking of them dry).
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

### Empty matrices

.An empty matrix is a matrix in which the number of rows or columns (or both) is zero.^ What the sight of the rows of numbers is meant to do is to remind us that Neo no longer believes in the Matrix illusion but understands it is a program, but even so, he should continue to see it.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ The way the word "matrix" refers both to the womb and to an array of numbers works perfectly.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[55][56] .An empty matrix has no entries but it still has a well defined number of rows and columns, which are needed for instance in the definition of the matrix product.^ What the sight of the rows of numbers is meant to do is to remind us that Neo no longer believes in the Matrix illusion but understands it is a program, but even so, he should continue to see it.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ He keeps everything running, and can produce maps of any nook and cranny of The Matrix as well as programs to produce any kind of expertise needed, such as the ability to fly a helicopter.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ All the human bio-electricity flows from the pod fields to the Source, and then it is redistributed back to the Matrix along well-defined channels.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

.Thus if A is the 3-by-0 matrix A and B is the 0-by-3 matrix B, then AB is the 3-by-3 zero matrix (corresponding to the null map from a 3-dimensional space V to itself obtained obtained as composition $g\circ f$ of the unique map f from V to a 0-dimensional space Z, followed by the zero map g from Z back to V), while BA is the 0-by-0 matrix (corresponding to the unique map from Z to itself obtained as composition $f\circ g$).^ The matrix that generates the matrix has described itself to its own self-generated programs as the "uncreated" -- thus giving form to yet another program ...
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

.There is no common notation for empty matrices but most computer algebra systems will allow creating them and computing with them.^ What you must learn is that these rules are no different that the rules of a computer system.
• The Matrix Transcript | Dialogue from the Movie 7 January 2010 23:54 UTC www.ix625.com [Source type: Original source]

^ So it makes no sense to think that a computer could be programmed with rules for producing the sensory-motor connections that would allow the creation of all possible worlds in advance of their being opened by human beings.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ The world that seems real to most people is in fact a computer-generated simulation, but almost no one knows it.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.Note that the determinant of the 0-by-0 matrix is 1 (and not 0 as might seem more natural): the Leibniz formula produces this value as a sum over the unique permutation of the empty set, with an empty product as term; also the Laplace expansion for a 1-by-1 matrix makes clear that the value of the 0-by-0 minor should be taken to be 1. This value is also consistent with the fact that the identity map from any finite dimensional space to itself has determinant 1, a fact that is often used as a part of the characterization of determinants.^ We should also note the perspective we have on the Matrix as viewers of The Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ I would like to use it as part of a cover for an ebook my husband and I are writing on “Breaking Free of the Matrix.” .
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ What makes you think you’re qualified to question whether this is a valuable use of my time?
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

## Applications

There are numerous applications of matrices, both in mathematics and other sciences. .Some of them merely take advantage of the compact representation of a set of numbers in a matrix.^ I can count the number of movies on one hand that take a thoughtful cybernetics viewpoint in constructing a story - the Matrix trilogy is one of the very few that do.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ True, the ones who see through the illusion of the Matrix can get over some of the limitations of having a body.23 But such flying takes place in the Matrix world.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ The urge to take the red pill remained, and a growing number of people refused to believe in the Matrix.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

.For example, in game theory and economics, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose.^ So in choosing to return from the "real world" to the Matrix world, Cypher is just choosing between two systematic sets of appearances.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ This duality between the matrix and the reality beyond it sets up the ultimate goal of the rebels, which is to free all minds from the matrix and allow humans to live out their lives in the real world beyond.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ In particular, he makes the case that, given a traditional Christian notion of an afterlife, Heaven turns out to be rather like a Matrix!
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[57] .Text mining and automated thesaurus compilation makes use of document-term matrices such as tf-idf in order to keep track of frequencies of certain words in several documents.^ In other words, compared with some easily imaginable possibilities, we are severely constrained, in a type of bondage, though ordinarily most of us don't think of it as such.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ The point of such an example is that words do not refer to objects "magically" or intrinsically: certain conditions must be met in the world in order for us to accept that a given written or spoken word has any meaning and whether it actually refers to anything at all.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[58]
Complex numbers can be represented by particular real 2-by-2 matrices via
$a + ib \leftrightarrow \begin{bmatrix} a & -b \ b & a \end{bmatrix},$
under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of absolute value 1, as above. A similar interpretation is possible for quaternions.[59]
Early encryption techniques such as the Hill cipher also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break.[60] .Computer graphics uses matrices both to represent objects and to calculate transformations of objects using affine rotation matrices to accomplish tasks such as projecting a three-dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation.^ Characters are stylized and symbolic, as they often are in dreams, representing some emotional pivot rather than a three-dimensional person (this is very obvious for the Agents).
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[61] Matrices over a polynomial ring are important in the study of control theory.
Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy. Examples are the overlap matrix and the Fock matrix using in solving the Roothaan equations to obtain the molecular orbitals of the Hartree–Fock method.

### Graph theory

An undirected graph with adjacency matrix $\begin{bmatrix} 2 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix}.$
The adjacency matrix of a finite graph is a basic notion of graph theory.[62] It saves which vertices of the graph are connected by an edge. .Matrices containing just two different values (0 and 1 meaning for example "yes" and "no") are called logical matrices.^ If these things are not "real" in the sense that their underlying constitution is radically other than I had believed, that makes no difference to the value that these things have in our lives.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Just to clarify, by no means do I reject a more holistic view of the trilogy.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ The location of Mobil Avenue station (I'll just call it Mobil from now on) can be very confusing, principally because, in the movie, it is described from two different perspectives at the same time.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

.The distance (or cost) matrix contains information about distances of the edges.^ The Matrix Online : Game Information About The Matrix Online for PC at MMORPG.COM Login: Password: Remember?
• The Matrix Online : Game Information About The Matrix Online for PC at MMORPG.COM 7 January 2010 23:54 UTC www.mmorpg.com [Source type: General]

[63] These concepts can be applied to websites connected hyperlinks or cities connected by roads etc., in which case (unless the road network is extremely dense) the matrices tend to be sparse, i.e. contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in network theory.

### Analysis and geometry

The Hessian matrix of a differentiable function ƒ: RnR consists of the second derivatives of ƒ with respect to the several coordinate directions, i.e.[64]
$H(f) = \left [\frac {\partial^2f}{\partial x_i \, \partial x_j} \right ].$
.It encodes information about the local growth behaviour of the function: given a critical point x = (x1, ..., xn), i.e., a point where the first partial derivatives $\partial f / \partial x_i$ of ƒ vanish, the function has a local minimum if the Hessian matrix is positive definite.^ Again, this was pointed out clearly in the first matrix.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ We have only bits and pieces of information but what we know for certain is that at some point in the early twenty-first century all of mankind was united in celebration.
• The Matrix: Script original 7 January 2010 23:54 UTC www.thematrixfr.com [Source type: Original source]

^ Given the deadpan delivery, it is hard to say whether it posits a deficiency in the machines that designed the Matrix, or in us in our notion of a perfect world.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above).^ Seeing the Oracle, and Sophia, as the archetype of the 'One' Soul, from which all individual souls emerge from the realm of matter, it is not the least bit surprising for me to find her in the kitchen baking cookies.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

[65]
At the saddle point (x = 0, y = 0) (red) of the function f(x,−y) = x2 − y2, the Hessian matrix $\begin{bmatrix} 2 & 0 \ 0 & -2 \end{bmatrix}$ is indefinite.
Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f: RnRm. If f1, ..., fm denote the components of f, then the Jacobi matrix is defined as [66]
$J(f) = \left [\frac {\partial f_i}{\partial x_j} \right ]_{1 \leq i \leq m, 1 \leq j \leq n}.$
If n > m, and if the rank of the Jacobi matrix attains its maximal value m, f is locally invertible at that point, by the implicit function theorem.[67]
Partial differential equations can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation. For elliptic partial differential equations this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question.[68]
The finite element method is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.[69]

### Probability theory and statistics

Two different Markov chains. The chart depicts the number of particles (of a total of 1000) in state "2". Both limiting values can be determined from the transition matrices, which are given by $\begin{bmatrix}.7&0\\.3&1\end{bmatrix}$ (red) and $\begin{bmatrix}.7&.2\\.3&.8\end{bmatrix}$ (black).
.Stochastic matrices are square matrices whose rows are probability vectors, i.e., whose entries sum up to one.^ It can be summed up with one shot...
• MATRIX REVOLUTIONS -- Ain't It Cool News: The best in movie, TV, DVD, and comic book news. 26 January 2010 0:55 UTC www.aintitcool.com [Source type: FILTERED WITH BAYES]

^ Everyone is hooked up to one and the same Matrix; there are not unique matrices generated for each individual.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

Stochastic matrices are used to define Markov chains with finitely many states.[70] .A row of the stochastic matrix gives the probability distribution for the next position of some particle which is currently in the state corresponding to the row.^ So I asked them if they would look for something in literature that represented some of the ideological themes that had influenced them when they were writing The Matrix that we could give to the choir and have them sing.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

Properties of the Markov chain like absorbing states, i.e. states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.[71]
Statistics also makes use of matrices in many different forms. .Descriptive statistics is concerned with describing data sets, which can often be represented in matrix form, by reducing the amount of data.^ The matrix that generates the matrix has described itself to its own self-generated programs as the "uncreated" -- thus giving form to yet another program ...
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

The covariance matrix encodes the mutual variance of several random variables.[72] Another technique using matrices are linear least squares, a method that approximates a finite set of pairs (x1, y1), (x2, y2), ..., (xN, yN), by a linear function
yiaxi + b, i = 1, ..., N
which can be formulated in terms of matrices, related to the singular value decomposition of matrices.[73]
Random matrices are matrices whose entries are random numbers, subject to suitable probability distributions, such as matrix normal distribution. Beyond probability theory, they are applied in domains ranging from number theory to physics.[74][75]

### Symmetries and transformations in physics

.Linear transformations and the associated symmetries play a key role in modern physics.^ The Oracle played a key part in Neo's transformation, a part that wouldn't have occurred.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

For example, elementary particles in quantum field theory are classified as representations of the Lorentz group of special relativity and, more specifically, by their behavior under the spin group. Concrete representations involving the Pauli matrices and more general gamma matrices are an integral part of the physical description of fermions, which behave as spinors.[76] .For the three lightest quarks, there is a group-theoretical representation involving the special unitary group SU(3); for their calculations, physicists use a convenient matrix representation known as the Gell-Mann matrices, which are also used for the SU(3) gauge group that forms the basis of the modern description of strong nuclear interactions, quantum chromodynamics.^ There is a small group "independent" humans who live outside the Matrix in place called Zion.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ ATP is the chemical humans use to store energy it has a link of three phosphate groups.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ In a sense, The Matrix is nothing more than a modern day "Big Brother," taking on a machine form rather than the Orwellian vision of a powerful individual using machines to assist and bring about an all-powerful status.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.The Cabibbo–Kobayashi–Maskawa matrix, in turn, expresses the fact that the basic quark states that are important for weak interactions are not the same as, but linearly related to the basic quark states that define particles with specific and distinct masses.^ The fact that one and the same Matrix is inhabited by millions of minds means that millions of people are really interacting, even if the physical universe in which they are interacting is radically different from how it appears.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ I am arguing above that the fact that one mind is really interacting with other minds is critical to assessing the value of the Matrix reality.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[77]

### Linear combinations of quantum states

The first model of quantum mechanics (Heisenberg, 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states.[78] This is also referred to as matrix mechanics. .One particular example is the density matrix that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" eigenstates.^ It is trying to put the audience in the same kind of state of mind as the inhabitants of the Matrix, so that we too are in our own Matrix_the one created by the filmmakers.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[79]
.Another matrix serves as a key tool for describing the scattering experiments which form the cornerstone of experimental particle physics: Collision reactions such as occur in particle accelerators, where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states.^ Thus, wrongly perceived situations may result in physical or psychological reactions.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ The pairs of opposites, the particle and the antiparticle, the yin and the yang of Neo, accelerate toward each other.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

^ The matrix that generates the matrix has described itself to its own self-generated programs as the "uncreated" -- thus giving form to yet another program ...
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

.The linear combination is given by a matrix known as the S-matrix, which encodes all information about the possible interactions between particles.^ With Sati's line we have all the information we need to fully construct what has happened with respect to the Matrix and the war.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

^ COMPLIMENTS I have always hated the fact that whenever I find some site to explain some of the doubts about Matrix, it starts telling me crap about the philosphy and all.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ One of the things that we had talked about when we first had the idea of The Matrix was an idea that I believe philosophy and religion and mathematics all try to answer.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

[80]

### Normal modes

A general application of matrices in physics is to the description of linearly coupled harmonic systems. .The equations of motion of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions.^ Neither the "stillness" of the pleroma nor the unchanging "nothingness" of nirvana are characterized by the dependence on technology and the use of force which so characterizes both of the worlds of the rebels in The Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ The matrix that generates the matrix has described itself to its own self-generated programs as the "uncreated" -- thus giving form to yet another program ...
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ In the context of the Matrix, you may see what you are permitted to see: the display that the system generates.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

The best way to obtain solutions is to determine the system's eigenvectors, its normal modes, by diagonalizing the matrix equation. Techniques like this are crucial when it comes to describing the internal dynamics of molecules: the internal vibrations of systems consisting of mutually bound component atoms.[81] They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.[82]

### Geometrical optics

Geometrical optics provides further matrix applications. In this approximative theory, the wave nature of light is neglected. The result is a model in which light rays are indeed geometrical rays. .If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix: the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element.^ There is a small group "independent" humans who live outside the Matrix in place called Zion.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ Compare two kinds of possible Matrix: the Second Matrix is communal, featuring real interaction between human beings call this human interaction; a solitary Matrix lacks human interaction altogether.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.Actually, there will be two different kinds of matrices, viz.^ However, there are two kinds of protection.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

a .refraction matrix describing de refraction at a lens surface, and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix will apply.^ The matrix that generates the matrix has described itself to its own self-generated programs as the "uncreated" -- thus giving form to yet another program ...
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ The films presentation of the matrix as a corporate network of human conceptions (or samsara) which are translated into software codes that reinforce one another illustrates this close relationship.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ The film's presentation of the Matrix as a corporate network of human conceptions (or samsara) which are translated into software codes that reinforce one another illustrates this close relationship.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

.The optical system consisting of a combination of lenses and/or reflective elements is simply described by the matrix resulting from the product of the components' matrices.^ Gnosticism in The Matrix [6] Although the presence of individual Christian elements within the film is clear, the overall system of Christianity that is presented is not the traditional, orthodox one.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ They are not the result of a systematic correlation between input and output to the brain's perceptual system that is meant to reproduce the consistent coordinated experience that we have when awake.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[83]

### Electronics

The behaviour of many electronic components can be described using matrices. Let A be a 2-dimensional vector with the component's input voltage v1 and input current i1 as its elements, and let B be a 2-dimensional vector with the component's output voltage v2 and output current i2 as its elements. .Then the behaviour of the electronic component can be described by B = H · A, where H is a 2 x 2 matrix containing one impedance element (h12), one admittance element (h21) and two dimensionless elements (h11 and h22).^ Morpheus opens a container which holds two pills : a blue one, and a red one.

^ The only aspect of your work I didn’t “like” was that you fell neccesarry that one should seperate the two interpretations of the Matrix Trilogy.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ Gnosticism in The Matrix [6] Although the presence of individual Christian elements within the film is clear, the overall system of Christianity that is presented is not the traditional, orthodox one.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

Calculating a circuit now reduces to multiplying matrices.

## History

Matrices have a long history of application in solving linear equations. The Chinese text The Nine Chapters on the Mathematical Art (Jiu Zhang Suan Shu), from between 300 BC and AD 200, is the first example of the use of matrix methods to solve simultaneous equations,[84] including the concept of determinants, almost 2000 years before its publication by the Japanese mathematician Seki in 1683 and the German mathematician Leibniz in 1693. Cramer presented Cramer's rule in 1750.
.Early matrix theory emphasized determinants more strongly than matrices and an independent matrix concept akin to the modern notion emerged only in 1858, with Cayley's Memoir on the theory of matrices.^ A more modern meaning of "matrix" is based in mathematics: a rectangular arrangement of symbols.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ In a sense, The Matrix is nothing more than a modern day "Big Brother," taking on a machine form rather than the Orwellian vision of a powerful individual using machines to assist and bring about an all-powerful status.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Thus, when Smith returns to the Matrix by taking over a battery person, he can make himself more powerful than an Agent.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

[85][86] .The term "matrix" was coined by Sylvester, who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices which derive from the original one by removing columns and rows.^ There is a small group "independent" humans who live outside the Matrix in place called Zion.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ One's self-esteem is determined by the group, even in so-called rebellion.
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ So who would have Seraph’s opposite been in that version of the matrix he existed in as the one?
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

.Etymologically, matrix derives from Latin mater (mother).^ "Matrix," is from the Latin for "mother," and originally meant "womb" (it is used in the Old Testament five times with this meaning), or "pregnant female."
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

[87]
The study of determinants sprang from several sources.[88] Number-theoretical problems led Gauss to relate coefficients of quadratic forms, i.e., expressions such as x2 + xy − 2y2, and linear maps in three dimensions to matrices. Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are non-commutative. Cauchy was the first to prove general statements about determinants, using as definition of the determinant of a matrix A = [ai,j] the following: replace the powers ajk by ajk in the polynomial
$a_1 a_2 \cdots a_n \prod_{i < j} (a_j - a_i)\;,$
where Π denotes the product of the indicated terms. He also showed, in 1829, that the eigenvalues of symmetric matrices are real.[89] Jacobi studied "functional determinants"—later called Jacobi determinants by Sylvester—which can be used to describe geometric transformations at a local (or infinitesimal) level, see above; Kronecker's Vorlesungen über die Theorie der Determinanten[90] and Weierstrass' Zur Determinantentheorie,[91] both published in 1903, first treated determinants axiomatically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established.
Many theorems were first established for small matrices only, for example the Cayley-Hamilton theorem was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by Hamilton for 4×4 matrices. Frobenius, working on bilinear forms, generalized the theorem to all dimensions (1898). Also at the end of the 19th century the Gauss-Jordan elimination (generalizing a special case now known as Gauss elimination) was established by Jordan. In the early 20th century, matrices attained a central role in linear algebra.[92]
The inception of matrix mechanics by Heisenberg, Born and Jordan led to studying matrices with infinitely many rows and columns.[93] Later, von Neumann carried out the mathematical formulation of quantum mechanics, by further developing functional analytic notions such as linear operators on Hilbert spaces, which, very roughly speaking, correspond to Euclidean space, but with an infinity of independent directions.

### Other historical usages of the word "matrix" in mathematics

The word has been used in unusual ways by at least two authors of historical importance.
Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910-1913) use the word matrix in the context of their Axiom of reducibility. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the "bottom" (0 order) the function will be identical to its extension:
"Let us give the name of matrix to any function, of however many variables, which does not involve any apparent variables. Then any possible function other than a matrix is derived from a matrix by means of generalization, i.e. by considering the proposition which asserts that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined".[94]
For example a function Φ(x, y) of two variables x and y can be reduced to a collection of functions of a single variable, e.g. y, by "considering" the function for all possible values of "individuals" ai substituted in place of variable x. And then the resulting collection of functions of the single variable y, i.e. ∀ai: Φ(ai, y), can be reduced to a "matrix" of values by "considering" the function for all possible values of "individuals" bi substituted in place of variable y:
∀bj∀ai: Φ(ai, bj).
Alfred Tarski in his 1946 Introduction to Logic used the word "matrix" synonymously with the notion of truth table as used in mathematical logic.[95]

## Notes

1. ^ Brown 1991, Chapter I.1. Alternative references for this book include Lang 1987b and Greub 1975.
2. ^ Oualline 2003, Ch. 5.
3. ^ Brown 1991, Definition I.2.1 (addition), Definition I.2.4 (scalar multiplication), and Definition I.2.33 (transpose)
4. ^ Brown 1991, Theorem I.2.6.
5. ^ Brown 1991, Definition I.2.20.
6. ^ Brown 1991, Theorem I.2.24.
7. ^ Horn & Johnson 1985, Ch. 4 and 5.
8. ^ Brown 1991, I.2.21 and 22.
9. ^ Greub 1975, Section III.2.
10. ^ Brown 1991, Definition II.3.3.
11. ^ Greub 1975, Section III.1.
12. ^ Brown 1991, Theorem II.3.22.
13. ^ Brown 1991, Definition I.2.28.
14. ^ Brown 1991, Definition I.5.13.
15. ^ This is immediate from the definition of matrix multiplication.
$\mathrm{tr}(AB) = \sum_{i=1}^m \sum_{j=1}^n A_{ij} B_{ji} = \mathrm{tr}(BA).$
16. ^ Brown 1991, Definition III.2.1.
17. ^ Brown 1991, Theorem III.2.12.
18. ^ Brown 1991, Corollary III.2.16.
19. ^ Mirsky 1990, Theorem 1.4.1.
20. ^ Brown 1991, Theorem III.3.18.
21. ^ Brown 1991, Definition III.4.1.
22. ^ Brown 1991, Definition III.4.9.
23. ^ Brown 1991, Corollary III.4.10.
24. ^ Horn & Johnson 1985, Theorem 2.5.6.
25. ^ Horn & Johnson 1985, Chapter 7.
26. ^ Horn & Johnson 1985, Theorem 7.2.1.
27. ^ Horn & Johnson 1985, Example 4.0.6, p. 169.
28. ^ Bau III & Trefethen 1997.
29. ^ Householder 1975, Ch. 7.
30. ^ Golub & Van Loan 1996, Algorithm 1.3.1.
31. ^ Golub & Van Loan 1996, Chapters 9 and 10, esp. section 10.2.
32. ^ Golub & Van Loan 1996, Chapter 2.3.
33. ^ For example, Mathematica, see Wolfram 2003, Ch. 3.7.
34. ^ Press, Flannery & Teukolsky 1992.
35. ^ Stoer & Bulirsch 2002, Section 4.1.
36. ^ Horn & Johnson 1985, Theorem 2.5.4.
37. ^ Horn & Johnson 1985, Ch. 3.1, 3.2.
38. ^ Arnold & Cooke 1992, Sections 14.5, 7, 8.
39. ^ Bronson 1989, Ch. 15.
40. ^ Coburn 1955, Ch. V.
41. ^ Lang 2002, Chapter XIII.
42. ^ Lang 2002, XVII.1, p. 643.
43. ^ Lang 2002, Proposition XIII.4.16.
44. ^ Reichl 2004, Section L.2.
45. ^ Greub 1975, Section III.3.
46. ^ Greub 1975, Section III.3.13.
47. ^ See any standard reference in group.
48. ^ Baker 2003, Def. 1.30.
49. ^ Baker 2003, Theorem 1.2.
50. ^ Artin 1991, Chapter 4.5.
51. ^ Artin 1991, Theorem 4.5.13.
52. ^ Rowen 2008, Example 19.2, p. 198.
53. ^ See any reference in representation theory or group representation.
54. ^ See the item "Matrix" in Itõ, ed. 1987.
55. ^ "Empty Matrix: A matrix is empty if either its row or column dimension is zero", Glossary, O-Matrix v6 User Guide
56. ^ "A matrix having at least one dimension equal to zero is called an empty matrix", MATLAB Data Structures
57. ^ Fudenberg & Tirole 1983, Section 1.1.1.
58. ^ Manning 1999, Section 15.3.4.
59. ^ Ward 1997, Ch. 2.8.
60. ^ Stinson 2005, Ch. 1.1.5 and 1.2.4.
61. ^ Association for Computing Machinery 1979, Ch. 7.
62. ^ Godsil & Royle 2004, Ch. 8.1.
63. ^ Punnen 2002.
64. ^ Lang 1987a, Ch. XVI.6.
65. ^ Nocedal 2006, Ch. 16.
66. ^ Lang 1987a, Ch. XVI.1.
67. ^ Lang 1987a, Ch. XVI.5. For a more advanced, and more general statement see Lang 1969, Ch. VI.2.
68. ^ Gilbarg & Trudinger 2001.
70. ^ Latouche & Ramaswami 1999.
71. ^ Mehata & Srinivasan 1978, Ch. 2.8.
72. ^ Krzanowski 1988, Ch. 2.2., p. 60.
73. ^ Krzanowski 1988, Ch. 4.1.
74. ^ Conrey 2007.
75. ^ Zabrodin, Brezin & Kazakov et al. 2006.
76. ^ Itzykson & Zuber 1980, Ch. 2.
77. ^ see Burgess & Moore 2007, section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi–Maskawa matrix).
78. ^ Schiff 1968, Ch. 6.
79. ^ Bohm 2001, sections II.4 and II.8.
80. ^ Weinberg 1995, Ch. 3.
81. ^ Wherrett 1987, part II.
82. ^ Riley, Hobson & Bence 1997, 7.17.
83. ^ Guenther 1990, Ch. 5.
84. ^ Shen, Crossley & Lun 1999 cited by Bretscher 2005, p. 1.
85. ^ Cayley 1889, vol. II, p. 475–496
86. ^ Dieudonné, ed. 1978, Vol. 1, Ch. III, p. 96
87. ^ Merriam-Webster dictionary, retrieved April, 20th 2009
88. ^ Knobloch 1994.
89. ^ Hawkins 1975.
90. ^ Kronecker 1897.
91. ^ Weierstrass 1915, pp. 271–286.
92. ^ Bôcher 2004.
93. ^ Mehra & Rechenberg 1987.
94. ^ Alfred North Whitehead and Bertrand Russell (1913) Principia Mathematica to *56, Cambridge at the University Press, Cambridge UK (republished 1962) cf page 162ff.
95. ^ Tarski, Alfred 1946 Introduction to Logic and the Methodology of Deductive Sciences, Dover Publications, Inc, New York NY, ISBN:0-486-28462-X.
1. ^ Eigen means "own" in German and in Dutch.
2. ^ Additionally, the group is required to be closed in the general linear group.
3. ^ "Not much of matrix theory carries over to infinite-dimensional spaces, and what does is not so useful, but it sometimes helps." Halmos 1982, p. 23, Chapter 5

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^ For authorized version of the script, see The Art of the Matrix , New Market Press, New York, 2000.

^ Elaine Pagels, The Gnostic Gospels, (New York: Random House, 1979, repr.
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• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

), .Prentice Hall
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• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ For authorized version of the script, see The Art of the Matrix , New Market Press, New York, 2000.

), .Berlin, New York: Springer-Verlag, ISBN 978-3-540-41160-4
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^ Buddhist Texts Through the Ages (New York: Philosophical Library, 1954), p.
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^ Buddhist Texts Through the Ages (New York: Philosophical Library, 1954), p.
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• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ Walter Kaufman (New York: Vintage Books, 1966).
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• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ For authorized version of the script, see The Art of the Matrix , New Market Press, New York, 2000.

(1987), Encyclopedic dictionary of mathematics. Vol. I--IV (2nd ed.), MIT Press, MR901762, ISBN 978-0-262-09026-1
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• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ Buddhist Texts Through the Ages (New York: Philosophical Library, 1954), p.
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• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

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^ His book, Natural Born Cyborg, is one of the best at penetrating this notion of the co-evolution of man and his tools, .
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ Bentley Layton, The Gnostic Scriptures (New York: Doubleday, 1995), Kurt Rudolph, Gnosis: The Nature and History of Gnosticism (San Francisco: HarperSanFrancisco, 1987).
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• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

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• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

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• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Reason, Truth, and History, Cambridge University Press, 1981.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

), .Berlin, New York: Springer-Verlag, ISBN 978-0-387-95452-3
• Ward, J. P. (1997), Quaternions and Cayley numbers, Mathematics and its Applications, 403, Dordrecht: Kluwer Academic Publishers Group, MR1458894, ISBN 978-0-7923-4513-8
• Wolfram, Stephen (2003), The Mathematica Book (5th ed.^ Walter Kaufman (New York: Vintage Books, 1966).
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Lowell Bair (New York: Bantam Books, 1961), 98.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

), Champaign, Ill: Wolfram Media, ISBN 978-1-57955-022-6

### Physics references

• Bohm, Arno (2001), Quantum Mechanics: Foundations and Applications, Springer, ISBN 0-387-95330-2
• Burgess, Cliff; Moore, Guy (2007), The Standard Model. .A Primer, Cambridge University Press, ISBN 0-521-86036-9
• Guenther, Robert D. (1990), Modern Optics, John Wiley, ISBN 0-471-60538-7
• Itzykson, Claude; Zuber, Jean-Bernard (1980), Quantum Field Theory, McGraw-Hill, ISBN 0-07-032071-3
• Riley, K. F.; Hobson, M. P.; Bence, S. J. (1997), Mathematical methods for physics and engineering, Cambridge University Press, ISBN 0-521-55506-X
• Schiff, Leonard I. (1968), Quantum Mechanics (3rd ed.^ The brothers explain, "There's something uniquely interesting about Buddhism and mathematics, particularly about quantum physics, and where they meet.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

^ Charles Taylor, "Overcoming Epistemology," Philosophical Arguments (Cambridge, MA: Harvard University Press, 1995), 12.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Cambridge University Press, 1984.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

), McGraw-Hill
• Weinberg, Steven (1995), The Quantum Theory of Fields. Volume I: Foundations, Cambridge University Press, ISBN 0-521-55001-7
• Wherrett, Brian S. (1987), Group Theory for Atoms, Molecules and Solids, Prentice-Hall International, ISBN 0-13-365461-3
• Zabrodin, Anton; Brezin, Édouard; Kazakov, Vladimir; Serban, Didina; Wiegmann, Paul (2006), Applications of Random Matrices in Physics (NATO Science Series II: Mathematics, Physics and Chemistry), Berlin, New York: Springer-Verlag, ISBN 978-1-4020-4530-1

### Historical references

.
• Bôcher, Maxime (2004), Introduction to higher algebra, New York: Dover Publications, ISBN 978-0-486-49570-5 , reprint of the 1907 original edition
•
• Dieudonné, Jean, ed.^ For authorized version of the script, see The Art of the Matrix , New Market Press, New York, 2000.

^ Cambridge University Press, 1984.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ Get a screensaver and view the rest of this Matrix site: http://www.whatisthematrix.com/cmp/screensaver_index.html .
• The Matrix: Reviews and Comments 7 January 2010 23:54 UTC www.nonduality.com [Source type: Original source]

(1978), Abrégé d'histoire des mathématiques 1700-1900, Paris: Hermann
• Hawkins, Thomas (1975), "Cauchy and the spectral theory of matrices", Historia Mathematica 2: 1–29, doi:10.1016/0315-0860(75)90032-4, MR0469635, ISSN 0315-0860
• Knobloch, Eberhard (1994), "From Gauss to Weierstrass: determinant theory and its historical evaluations", The intersection of history and mathematics, Sci. Networks Hist. Stud., 15, Basel, Boston, Berlin: Birkhäuser, pp. 51–66, MR1308079
•
• Mehra, J.; Rechenberg, Helmut (1987), The Historical Development of Quantum Theory (1st ed.), .Berlin, New York: Springer-Verlag, ISBN 978-0-387-96284-9
• Shen, Kangshen; Crossley, John N.; Lun, Anthony Wah-Cheung (1999), Nine Chapters of the Mathematical Art, Companion and Commentary (2nd ed.^ For authorized version of the script, see The Art of the Matrix , New Market Press, New York, 2000.

), Oxford University Press, ISBN 978-0-19-853936-0
•

History
Online books
Online matrix calculators
Example Source Code

# Study guide

Up to date as of January 14, 2010

### From Wikiversity

A matrix is a two dimensional array. An array is a list of data values. .A matrix, or two dimensional array is a list of values that cross two axes and would appear, conceptually, as a grid.^ If two people fall in love in the Matrix, in what sense would their love not be real?
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ If the original Matrix film had been the only story -- if the second two movies had not been made -- this would be absolutely correct.
• The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]

^ The Matrix would appear to be more morally responsible to its human subjects than are human subjects to themselves.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

A multiplication table could be an example of a matrix.

## Data processing

.In Perl for example, a matrix is identified usually as a hash table called, for instance, %Matrix and can be used to identify and define a set of database references.^ Since we'll be discussing different kinds of Matrix, we need a name for the one depicted in The Matrix; Agent Smith refers to a First Matrix, so let's call the one we see the Second Matrix.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

^ So, is that the child is (1) use his mind to change the Matrix ’spoon’ object’s property (in programming sense) (2) Neo is reference to the same ’spoon’ object (3) Neo saw it bend.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ For instance, much to my great dissapointment, the commentaries on the Matrix box set, for instance, barely even mention the idea that the Matrix trilogy is also a SciFi movie.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

.Both two-dimensional and three-dimensional matrices can be constructed where a list of lists or referential arrays are referenced using "key-value pairs". This can be accomplished in a standard way using a number of programming languages.^ This is analogous to the different ways of thinking about Putnam's Twin Earth scenario, common in discussions of two-dimensional semantics.
• The Matrix as Metaphysics 7 January 2010 23:54 UTC consc.net [Source type: Original source]

^ The vagueness AGAIN comes from the issue that there are two separate storylines going on here - the scene, as with most key scenes, has to accomodate both of them.
• Cyberpunk Review » The Matrix Trilogy: A Man-Machine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]

^ The way the word "matrix" refers both to the womb and to an array of numbers works perfectly.
• The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]

# Simple English

In mathematics, a matrix (plural matrices) is a rectangular table of numbers. There are rules for adding, subtracting and "multiplying" matrices together, but the rules are different than for numbers. As an example, $A \cdot B$ does not always give the same result as $B \cdot A$, which is the case for the multiplication of ordinary numbers.

Many natural sciences use matrices quite a lot. In many universities, courses about matrices (usually called linear algebra) are taught very early, sometimes even in the first year of studies. Matrices are also very common in computer science.

## = Definitions and notations

= The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called an m-by-n matrix (or m×n matrix) and m and n are called its dimensions.

The places in the matrix where the numbers are, are called entries. The entry of a matrix A that lies in the row number i and column number j is called the i,j entry of A. This is written as A[i,j] or aij.

We write $A:=\left(a_\left\{ij\right\}\right)_\left\{m \times n\right\}$ to define an m × n matrix A with each entry in the matrix called aij for all 1 ≤ im and 1 ≤ jn.

#### Example

The matrix

$\begin\left\{bmatrix\right\}$

1 & 2 & 3 \\ 1 & 2 & 7 \\ 4 & 9 & 2 \\ 6 & 1 & 5 \end{bmatrix}

is a 4×3 matrix. This matrix has m=4 rows, and n=3 columns.

The element A[2,3] or a23 is 7.

### Operations

The sum of two matrices is the matrix, which (i,j)-th entry is equal to the sum of the (i,j)-th entries of two matrices:


 \begin{bmatrix}
1 & 3 & 2 \\
1 & 0 & 0 \\
1 & 2 & 2
\end{bmatrix}
+
\begin{bmatrix}
0 & 0 & 5 \\
7 & 5 & 0 \\
2 & 1 & 1
\end{bmatrix}
=
\begin{bmatrix}
1+0 & 3+0 & 2+5 \\
1+7 & 0+5 & 0+0 \\
1+2 & 2+1 & 2+1
\end{bmatrix}
=
\begin{bmatrix}
1 & 3 & 7 \\
8 & 5 & 0 \\
3 & 3 & 3
\end{bmatrix}


The two matrices have the same dimensions. Here $A + B = B + A$ is true.

#### Multiplication of two matrices

The multiplication of two matrices is a bit more complicated:


 \begin{bmatrix}
a1 & a2  \\
a3 & a4 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
b1 & b2 \\
b3 & b4 \\
\end{bmatrix}
=
\begin{bmatrix}
(a1\cdot b1  +  a2 \cdot b3) &
(a1 \cdot b2 +  a2 \cdot b4) \\
(a3\cdot b1  +  a4 \cdot b3) &
(a3 \cdot b2 +  a4 \cdot b4) \\
\end{bmatrix}


So with Numbers:


 \begin{bmatrix}
3 & 5  \\
1 & 4 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
2 & 3 \\
5 & 0 \\
\end{bmatrix}
=
\begin{bmatrix}
(3\cdot 2  +  5 \cdot 5) &
(3 \cdot 3 +  5 \cdot 0) \\
(1\cdot 2  +  4 \cdot 5) &
(1 \cdot 3 +  4 \cdot 0) \\
\end{bmatrix}


=

\begin{bmatrix}
31 & 9 \\
22 & 3 \\
\end{bmatrix}


• two matrices can be multiplied with each other even if they have different dimensions, as long as the number of columns in the first matrix is equal to the number of rows in the second matrix.
• the result of the multiplication, called the product, is another matrix with the same number of rows as the first matrix and the same number of columns as the second matrix.
• the multiplication of matrices is not commutative, this means, in general that$A \cdot B \neq B \cdot A$
• the multiplication of matrices is associative, this means $\left(A \cdot B\right)\cdot C = A\cdot\left(B\cdot C\right)$

### Special matrices

There are some matrices that are special.

#### Square matrix

A square matrix has the same number of rows as columns, so m=n.

An example of a square matrix is

$\begin\left\{bmatrix\right\}$
5 & -2 & 4 \\
0 &  9 & 1 \\


-7 & 6 & 8 \\ \end{bmatrix}

This matrix has 3 rows and 3 columns: m=n=3.

#### Identity

Every square dimension set of a matrix has a special counterpart called an "identity matrix". The identity matrix has nothing but zeroes except on the main diagonal, where there are all ones. For example:

$\begin\left\{bmatrix\right\}$
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\


\end{bmatrix} is an identity matrix. There is exactly one identity matrix for each square dimension set. An identity matrix is special because when multiplying any matrix by the identity matrix, the result is always the original matrix with no change.

#### Inverse matrix

An inverse matrix is a matrix that, when multiplied by another matrix, equals the identity matrix. For example:

$\begin\left\{bmatrix\right\}$
7 & 8 \\
6 & 7 \\


\end{bmatrix} \cdot \begin{bmatrix}

7 & -8 \\
-6 & 7 \\


\end{bmatrix} = \begin{bmatrix}

1 & 0 \\
0 & 1 \\


\end{bmatrix}
$\begin\left\{bmatrix\right\}$

7 & -8 \\
-6 & 7 \\


\end{bmatrix} is the inverse of $\begin\left\{bmatrix\right\}$

7 & 8 \\
6 & 7 \\


\end{bmatrix}.

## One column matrix

A matrix, that has many rows, but only one column, is called a column vector.

## Determinants

The determinant takes a square matrix and returns a number. To understand what the number means, take each column of the matrix and draw it as a vector. The parallelogram drawn by those vectors has an area, which is the determinant. For all 2x2 matrices, the formula is very simple: $det\left\left( \begin\left\{bmatrix\right\} a & b \\ c & d \\ \end\left\{bmatrix\right\}\right\right) =ad - bc$

For 3x3 matrices the formula is more complicated: $det\left\left( \begin\left\{bmatrix\right\} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end\left\{bmatrix\right\}\right\right) = a_1\left(b_2 c_3 - c_2 b_3\right) - a_2\left(b_1 c_3 - c_1 b_3\right) + a_3\left(b_1 c_2 - c_1 b_2\right)$

There are no simple formulas for the determinants of larger matrices, and many computer programmers study how to get computers to quickly find large determinants.

### Properties of determinants

There are three rules that all determinants follow. These are:

• The determinant of an identity matrix is 1
• If two rows or two columns of the matrix are exchanged, then the determinant is multiplied by -1. Mathematicians call this alternating.
• If all the numbers in one row or column are multiplied by another number n, then the determinant is multiplied by n. Also, if a matrix M has a column v that is the sum of two column matrices $v_1$ and $v_2$, then the determinant of M is the sum of the determinants of M with $v_1$ in place of v and M with $v_2$ in place of v. These two conditions are called multi-linearity.

# Citable sentences

Up to date as of December 12, 2010

Here are sentences from other pages on The Matrix, which are similar to those in the above article.