From Wikipedia, the free 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

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.
.^ 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.
.^ 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.
.^ 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.
.^ 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.
.^ First principle: any abstract computation that could be used to simulate physical spacetime 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]
.^ 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.
.^ 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 sensorymotor 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 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,
.^ 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
mby
n matrix or
m ×
n matrix, while
m and
n are called its
dimensions. The above is a 4by3 matrix.
.^ 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]
.^ 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
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
uppercase letters, while the corresponding
lowercase letters, with two subscript indices, represent the entries.
.^ 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 doubleunderline with the variable name, with or without boldface style, (e.g.,
).
The entry that lies in the ith row and the jth 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 a_{i,j}. Alternative notations for that entry are A[i,j] or A_{i,j}.
.^ 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 ManMachine 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 ManMachine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]
For example,
a_{i,∗} refers to the i
^{th} row of
A, and
a_{∗,j} refers to the j
^{th} column of
A. The set of all
mby
n matrices is denoted
M(
m,
n).
A common shorthand is
 A = [a_{i,j}]_{i=1,...,m; j=1,...,n} or more briefly A = [a_{i,j}]_{m×n}
to define an m × n matrix A. Usually the entries a_{i,j} are defined separately for all integers 1 ≤ i ≤ m and 1 ≤ j ≤ n. They can however sometimes be given by one formula; for example the 3by4 matrix
can alternatively be specified by A = [i − j]_{i=1,2,3; j=1,...,4}.
.^ 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 ManMachine 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 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 mbyn matrices A and B is calculated entrywise:
 (A + B)_{i,j} = A_{i,j} + B_{i,j}, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.


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 · A_{i,j}.


Transpose 
The transpose of an mbyn matrix A is the nbym matrix A^{T} (also denoted A^{tr} or ^{t}A) formed by turning rows into columns and vice versa:
 (A^{T})_{i,j} = A_{j,i}.


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(
A^{T}) and (
A +
B)
^{T} =
A^{T} +
B^{T}. Finally, (
A^{T})
^{T} =
A.
Row operations are ways to change matrices.
.^ There are MANY people asking LOTS of “what about” type questions for the matrix movies. Cyberpunk Review » The Matrix Trilogy: A ManMachine 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!
.^ 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.
.^ 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 ManMachine 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
mby
n matrix and
B is an
nby
p matrix, then their
matrix product AB is the
mby
p matrix whose entries are given by dotproduct of the corresponding row of
A and the corresponding column of
B:
where 1 ≤ i ≤ m and 1 ≤ j ≤ p.^{[5]} For example (the underlined entry 1 in the product is calculated as the product 1 · 1 + 0 · 1 + 2 · 0 = 1):
.^ 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
mby
n and
nby
k matrices, respectively, and
m ≠ k. Even if both products are defined, they need not be equal, i.e. generally one has
 AB ≠ BA,
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:
whereas
.^ 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]
.^ 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]
.^ 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]} .^ 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×1matrix) of n variables x_{1}, x_{2}, ..., x_{n}, and A is an mbyn matrix, then the matrix equation
 Ax = b,
where b is some m×1column vector, is equivalent to the system of linear equations
 A_{1,1}x_{1} + A_{1,2}x_{2} + ... + A_{1,n}x_{n} = b_{1}
 ...
 A_{m,1}x_{1} + A_{m,2}x_{2} + ... + A_{m,n}x_{n} = b_{m} .^{[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.
.^ 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:
R^{n} →
R^{m} arises from a unique
mby
n matrix
A: explicitly, the
(i, j)entry of
A is the
i^{th} coordinate of
f(
e_{j}), where
e_{j} = (0,...,0,1,0,...,0) is the
unit vector with 1 in the
j^{th} 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
2by2 matrices with the associated linear maps of
R^{2}. The blue original is mapped to the green grid and shapes, the origin (0,0) is marked with a black point.
Under the
1to1 correspondence between matrices and linear maps, matrix multiplication corresponds to
composition of maps
^{[9]}: if a
kby
m matrix
B represents another linear map
g :
R^{m} →
R^{k}, then the composition
g ∘ f is represented by
BA since
 (g ∘ f)(x) = g(f(x)) = g(Ax) = B(Ax) = (BA)x.
.^ 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]
.^ 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]} .^ 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
.^ 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]
.^ 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
nonsingular if there exists a matrix
B such that
 AB = I_{n}.^{[13]}
This is equivalent to
BA =
I_{n}.
^{[14]} .^ It exists now, only as part of a neuralinteractive 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 neuralinteractive 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 A
_{i,i} form the
main diagonal of a matrix. The
trace, tr(
A) of a square matrix
A is the sum of its diagonal entries.
.^ 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]}
.^ 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]
.^ 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
 (diagonal), (lower) and (upper triangular matrix).
Determinant
Main article:
Determinant
A linear transformation on
R^{2} 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
R^{2}) or volume (in
R^{3}) 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 2by2 matrices is given by
the determinant of 3by3 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.
.^ 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 ManMachine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]
^ These two aspects – his sysadmin 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 ManMachine 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]} .^ 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 mindhealing 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 nonzero 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×
nmatrix
A if and only if
A−λ
I_{n} is not invertible, which is
equivalent to
 det(A−λI) = 0.^{[22]}
The function
p_{A}(
t) = det(
A−
tI) is called the
characteristic polynomial of
A, its
degree is
n. Therefore
p_{A}(
t) has at most
n different roots, i.e., eigenvalues of the matrix.
^{[23]} .^ 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
CayleyHamilton theorem,
p_{A}(A) = 0, that is to say, the characteristic polynomial applied to the matrix itself yields the
zero matrix.
Symmetry
.^ 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 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 infinitedimensional situations related to matrices with infinitely many rows and columns, see
below.
Definiteness
Matrix A; definiteness; associated quadratic form Q_{A}(x,y);
set of vectors (x,y) such that Q_{A}(x,y)=1 


positive definite 
indefinite 
1/4 x^{2} + y^{2} 
1/4 x^{2} − 1/4 y^{2} 
Ellipse 
Hyperbola 
A symmetric
n×
nmatrix is called
positivedefinite (respectively negativedefinite; indefinite), if for all nonzero vectors
x ∈
R^{n} the associated
quadratic form given by
 Q(x) = x^{T}Ax
takes only 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]} .^ It exists now, only as part of a neuralinteractive 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 neuralinteractive 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]
.^ 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 2by2 matrices.
Allowing as input two different vectors instead yields the
bilinear form associated to
A:
 B_{A} (x, y) = x^{T}Ay.^{[27]}
Computational aspects
.^ 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.
.^ 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
x_{n} 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.
.^ 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
n^{2.807} multiplications.
^{[30]} A refined approach also incorporates specific features of the computing devices.
.^ 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 ManMachine 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]}
.^ 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)
 A^{−1} = Adj(A) / det(A)
may lead to significant rounding errors if the determinant of the matrix is very small.
.^ 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]}
.^ 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 sensorymotor 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]
.^ 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 ManMachine 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
.^ 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.
.^ 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 ManMachine 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]} .^ 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 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 films 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.
.^ "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 selfrealized, 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 ManMachine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]
^{[37]} Given the eigendecomposition, the
n^{th} power of
A (i.e.
nfold iterated matrix multiplication) can be calculated via
 A^{n} = (VDV^{−1})^{n} = VDV^{−1}VDV^{−1}...VDV^{−1} = VD^{n}V^{−1}
Abstract algebraic aspects and generalizations
Matrices can be generalized in different ways.
.^ 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 ManMachine 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 higherdimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or twodimensional 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 wellbehaved, 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.
.^ 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 ManMachine 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]
.^ 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]
.^ Your analysis is the only one which considers it from a scientific point of view. Cyberpunk Review » The Matrix Trilogy: A ManMachine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]
.^ 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 ManMachine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]
^{[41]} .^ The human body generates more bioelectricity than a 120volt 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 T800 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 bioelectricity than a 120volt battery and over 25,000 BTUs of bodyheat. 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.
.^ 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.
.^ 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 ManMachine 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]}
.^ 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]
.^ 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
R^{n} →
R^{m} are equivalent to
mby
n matrices, as described
above. More generally, any linear map
f: V → W between finite
dimensional vector spaces can be described by a matrix
A = (
a_{ij}), after choosing
bases v_{1}, ...,
v_{n} of
V, and
w_{1}, ...,
w_{m} of
W (so
n is the dimension of
V and
m is the dimension of
W), which is such that
.^ In his reentry into the matrix, however, Neo retains the "residual selfimage" 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 scifi 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 ManMachine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]
^ In his reentry into the Matrix, however, Neo retains the "residual selfimage" 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 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
Main article:
Matrix group
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 computergenerated 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]
.^ 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 ManMachine 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
 M^{T}M = I,
form the
orthogonal group.
^{[50]} They are called
orthogonal since the associated linear transformations of
R^{n} preserve angles in the sense that the
scalar product of two vectors is unchanged after applying
M to them:
 (Mv) · (Mw) = v · w.^{[51]}
.^ 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 ManMachine 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 antione 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]} .^ By “using his powers outside the matrix” I mean that he was able to “exist” outside the matrix. Cyberpunk Review » The Matrix Trilogy: A ManMachine 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 computergenerated 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
.^ 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]
.^ 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 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 reentry into the matrix, however, Neo retains the "residual selfimage" 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 reentry into the Matrix, however, Neo retains the "residual selfimage" 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]
.^ 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]
.^ 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 ManMachine 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.
.^ 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]
.^ 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 columnindex and rowindex 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.
.^ If one takes the metaphysical view, Neo becomes selfrealized, 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 ManMachine 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 ManMachine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]
Empty matrices
.^ 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]} .^ 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 bioelectricity flows from the pod fields to the Source, and then it is redistributed back to the Matrix along welldefined channels. 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 selfgenerated 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]
.^ 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 sensorymotor 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 computergenerated simulation, but almost no one knows it. The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]
.^ 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 ManMachine 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 ManMachine 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.
.^ 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 ManMachine 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]
.^ 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]} .^ 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 2by2 matrices via
under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2by2 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]} .^ Characters are stylized and symbolic, as they often are in dreams, representing some emotional pivot rather than a threedimensional 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.
Graph theory
An undirected graph with adjacency matrix
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.
.^ 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 ManMachine 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 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
.^ Again, this was pointed out clearly in the first matrix. Cyberpunk Review » The Matrix Trilogy: A ManMachine 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 twentyfirst 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]
.^ 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) =
x^{2} −
y^{2}, the Hessian matrix
is
indefinite.
Another matrix frequently used in geometrical situations is the
Jacobi matrix of a differentiable map
f:
R^{n} →
R^{m}. If
f_{1}, ...,
f_{m} denote the components of
f, then the Jacobi matrix is defined as
^{[66]}
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]}
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
(red) and
(black).
.^ 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]} .^ 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.
.^ The matrix that generates the matrix has described itself to its own selfgenerated 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 (
x_{1},
y_{1}), (
x_{2},
y_{2}), ..., (
x_{N},
y_{N}), by a linear function
 y_{i} ≈ ax_{i} + b, i = 1, ..., N
Symmetries and transformations in 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]} .^ 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 ManMachine 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 allpowerful status. The Philosophy of the Matrix 7 January 2010 23:54 UTC onwardoverland.com [Source type: Original source]
.^ 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 infinitedimensional matrices acting on quantum states.
^{[78]} This is also referred to as
matrix mechanics.
.^ 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]}
.^ 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 selfgenerated 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]
.^ 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 ManMachine 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.
.^ 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 selfgenerated 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.
.^ 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]
.^ However, there are two kinds of protection. The Matrix: Revolutions, Explained 7 January 2010 23:54 UTC wylfing.net [Source type: Original source]
a
.^ The matrix that generates the matrix has described itself to its own selfgenerated 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 films 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]
.^ 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 2dimensional vector with the component's input voltage
v_{1} and input current
i_{1} as its elements, and let
B be a 2dimensional vector with the component's output voltage
v_{2} and output current
i_{2} as its elements.
.^ 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 ManMachine 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
.^ 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 allpowerful 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 ManMachine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]
^{[85]}^{[86]} .^ 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 selfesteem is determined by the group, even in socalled 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 ManMachine Interface Perspective 26 January 2010 0:55 UTC www.cyberpunkreview.com [Source type: FILTERED WITH BAYES]
.^ "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]} Numbertheoretical problems led Gauss to relate coefficients of
quadratic forms, i.e., expressions such as
x^{2} + xy − 2y^{2}, and
linear maps in three dimensions to matrices.
Eisenstein further developed these notions, including the remark that, in modern parlance,
matrix products are
noncommutative.
Cauchy was the first to prove general statements about determinants, using as definition of the determinant of a matrix
A = [
a_{i,j}] the following: replace the powers
a_{j}^{k} by
a_{jk} in the
polynomial
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
CayleyHamilton 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
GaussJordan 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]}
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.
 "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" a_{i} substituted in place of variable x. And then the resulting collection of functions of the single variable y, i.e. ∀a_{i}: Φ(a_{i}, y), can be reduced to a "matrix" of values by "considering" the function for all possible values of "individuals" b_{i} substituted in place of variable y:
 ∀b_{j}∀a_{i}: Φ(a_{i}, b_{j}).
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]}
See also
Notes
 ^ Brown 1991, Chapter I.1. Alternative references for this book include Lang 1987b and Greub 1975.
 ^ Oualline 2003, Ch. 5.
 ^ Brown 1991, Definition I.2.1 (addition), Definition I.2.4 (scalar multiplication), and Definition I.2.33 (transpose)
 ^ Brown 1991, Theorem I.2.6.
 ^ Brown 1991, Definition I.2.20.
 ^ Brown 1991, Theorem I.2.24.
 ^ Horn & Johnson 1985, Ch. 4 and 5.
 ^ Brown 1991, I.2.21 and 22.
 ^ Greub 1975, Section III.2.
 ^ Brown 1991, Definition II.3.3.
 ^ Greub 1975, Section III.1.
 ^ Brown 1991, Theorem II.3.22.
 ^ Brown 1991, Definition I.2.28.
 ^ Brown 1991, Definition I.5.13.
 ^ This is immediate from the definition of matrix multiplication.
 ^ Brown 1991, Definition III.2.1.
 ^ Brown 1991, Theorem III.2.12.
 ^ Brown 1991, Corollary III.2.16.
 ^ Mirsky 1990, Theorem 1.4.1.
 ^ Brown 1991, Theorem III.3.18.
 ^ Brown 1991, Definition III.4.1.
 ^ Brown 1991, Definition III.4.9.
 ^ Brown 1991, Corollary III.4.10.
 ^ Horn & Johnson 1985, Theorem 2.5.6.
 ^ Horn & Johnson 1985, Chapter 7.
 ^ Horn & Johnson 1985, Theorem 7.2.1.
 ^ Horn & Johnson 1985, Example 4.0.6, p. 169.
 ^ Bau III & Trefethen 1997.
 ^ Householder 1975, Ch. 7.
 ^ Golub & Van Loan 1996, Algorithm 1.3.1.
 ^ Golub & Van Loan 1996, Chapters 9 and 10, esp. section 10.2.
 ^ Golub & Van Loan 1996, Chapter 2.3.
 ^ For example, Mathematica, see Wolfram 2003, Ch. 3.7.
 ^ Press, Flannery & Teukolsky 1992.
 ^ Stoer & Bulirsch 2002, Section 4.1.
 ^ Horn & Johnson 1985, Theorem 2.5.4.
 ^ Horn & Johnson 1985, Ch. 3.1, 3.2.
 ^ Arnold & Cooke 1992, Sections 14.5, 7, 8.
 ^ Bronson 1989, Ch. 15.
 ^ Coburn 1955, Ch. V.
 ^ Lang 2002, Chapter XIII.
 ^ Lang 2002, XVII.1, p. 643.
 ^ Lang 2002, Proposition XIII.4.16.
 ^ Reichl 2004, Section L.2.
 ^ Greub 1975, Section III.3.
 ^ Greub 1975, Section III.3.13.
 ^ See any standard reference in group.
 ^ Baker 2003, Def. 1.30.
 ^ Baker 2003, Theorem 1.2.
 ^ Artin 1991, Chapter 4.5.
 ^ Artin 1991, Theorem 4.5.13.
 ^ Rowen 2008, Example 19.2, p. 198.
 ^ See any reference in representation theory or group representation.
 ^ See the item "Matrix" in Itõ, ed. 1987.
 ^ "Empty Matrix: A matrix is empty if either its row or column dimension is zero", Glossary, OMatrix v6 User Guide
 ^ "A matrix having at least one dimension equal to zero is called an empty matrix", MATLAB Data Structures
 ^ Fudenberg & Tirole 1983, Section 1.1.1.
 ^ Manning 1999, Section 15.3.4.
 ^ Ward 1997, Ch. 2.8.
 ^ Stinson 2005, Ch. 1.1.5 and 1.2.4.
 ^ Association for Computing Machinery 1979, Ch. 7.
 ^ Godsil & Royle 2004, Ch. 8.1.
 ^ Punnen 2002.
 ^ Lang 1987a, Ch. XVI.6.
 ^ Nocedal 2006, Ch. 16.
 ^ Lang 1987a, Ch. XVI.1.
 ^ Lang 1987a, Ch. XVI.5. For a more advanced, and more general statement see Lang 1969, Ch. VI.2.
 ^ Gilbarg & Trudinger 2001.
 ^ Šolin 2005, Ch. 2.5. See also stiffness method.
 ^ Latouche & Ramaswami 1999.
 ^ Mehata & Srinivasan 1978, Ch. 2.8.
 ^ Krzanowski 1988, Ch. 2.2., p. 60.
 ^ Krzanowski 1988, Ch. 4.1.
 ^ Conrey 2007.
 ^ Zabrodin, Brezin & Kazakov et al. 2006.
 ^ Itzykson & Zuber 1980, Ch. 2.
 ^ see Burgess & Moore 2007, section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi–Maskawa matrix).
 ^ Schiff 1968, Ch. 6.
 ^ Bohm 2001, sections II.4 and II.8.
 ^ Weinberg 1995, Ch. 3.
 ^ Wherrett 1987, part II.
 ^ Riley, Hobson & Bence 1997, 7.17.
 ^ Guenther 1990, Ch. 5.
 ^ Shen, Crossley & Lun 1999 cited by Bretscher 2005, p. 1.
 ^ Cayley 1889, vol. II, p. 475–496
 ^ Dieudonné, ed. 1978, Vol. 1, Ch. III, p. 96
 ^ MerriamWebster dictionary, http://www.merriamwebster.com/dictionary/matrix, retrieved April, 20th 2009
 ^ Knobloch 1994.
 ^ Hawkins 1975.
 ^ Kronecker 1897.
 ^ Weierstrass 1915, pp. 271–286.
 ^ Bôcher 2004.
 ^ Mehra & Rechenberg 1987.
 ^ Alfred North Whitehead and Bertrand Russell (1913) Principia Mathematica to *56, Cambridge at the University Press, Cambridge UK (republished 1962) cf page 162ff.
 ^ Tarski, Alfred 1946 Introduction to Logic and the Methodology of Deductive Sciences, Dover Publications, Inc, New York NY, ISBN:048628462X.
 ^ Eigen means "own" in German and in Dutch.
 ^ Additionally, the group is required to be closed in the general linear group.
 ^ "Not much of matrix theory carries over to infinitedimensional spaces, and what does is not so useful, but it sometimes helps." Halmos 1982, p. 23, Chapter 5
References
.