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Study guide

Up to date as of January 14, 2010

From Wikiversity

Much of finite elements revolves around forming matrices and solving systems of linear equations using matrices. This learning resource gives you a brief review of matrices.

Contents

Matrices

Suppose that you have a linear system of equations

 \begin{align} a_{11} x_1 + a_{12} x_2 + a_{13} x_3 + a_{14} x_4 &= b_1 \ a_{21} x_1 + a_{22} x_2 + a_{23} x_3 + a_{24} x_4 &= b_2 \ a_{31} x_1 + a_{32} x_2 + a_{33} x_3 + a_{34} x_4 &= b_3 \ a_{41} x_1 + a_{42} x_2 + a_{43} x_3 + a_{44} x_4 &= b_4 \end{align} ~.

Matrices provide a simple way of expressing these equations. Thus, we can instead write

 \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \ a_{21} & a_{22} & a_{23} & a_{24} \ a_{31} & a_{32} & a_{33} & a_{34} \ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \\ b_4 \end{bmatrix} ~.

An even more compact notation is

 \left[\mathsf{A}\right] \left[\mathsf{x}\right] = \left[\mathsf{b}\right] ~~~~\text{or}~~~~ \mathbf{A} \mathbf{x} = \mathbf{b} ~.

Here \mathbf{A} is a 4\times 4 matrix while \mathbf{x} and \mathbf{b} are 4\times 1 matrices. In general, an m \times n matrix \mathbf{A} is a set of numbers arranged in m rows and n columns.

 \mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \dots & a_{1n} \ a_{21} & a_{22} & a_{23} & \dots & a_{2n} \ \vdots & \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & a_{m3} & \dots & a_{mn} \end{bmatrix}~.

Types of Matrices

Common types of matrices that we encounter in finite elements are:

  • a row vector that has one row and n columns.
 \mathbf{v} = \begin{bmatrix} v_1 & v_2 & v_3 & \dots & v_n \end{bmatrix}
  • a column vector that has n rows and one column.
 \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ \vdots \\ v_n \end{bmatrix}
  • a square matrix that has an equal number of rows and columns.
  • a diagonal matrix which is a square matrix with only the

diagonal elements (aii) nonzero.

 \mathbf{A} = \begin{bmatrix} a_{11} & 0 & 0 & \dots & 0 \ 0 & a_{22} & 0 & \dots & 0 \ \vdots & \vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & \dots & a_{nn} \end{bmatrix}~.
  • the identity matrix (\mathbf{I}) which is a diagonal matrix and

with each of its nonzero elements (aii) equal to 1.

 \mathbf{A} = \begin{bmatrix} 1 & 0 & 0 & \dots & 0 \ 0 & 1 & 0 & \dots & 0 \ \vdots & \vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & \dots & 1 \end{bmatrix}~.
  • a symmetric matrix which is a square matrix with elements

such that aij = aji.

 \mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \dots & a_{1n} \ a_{12} & a_{22} & a_{23} & \dots & a_{2n} \ a_{13} & a_{23} & a_{33} & \dots & a_{3n} \ \vdots & \vdots & \vdots & \ddots & \vdots \ a_{1n} & a_{2n} & a_{3n} & \dots & a_{nn} \end{bmatrix}~.
  • a skew-symmetric matrix which is a square matrix with elements

such that aij = − aji.

 \mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \dots & a_{1n} \ -a_{12} & a_{22} & a_{23} & \dots & a_{2n} \ -a_{13} & -a_{23} & a_{33} & \dots & a_{3n} \ \vdots & \vdots & \vdots & \ddots & \vdots \ -a_{1n} & -a_{2n} & -a_{3n} & \dots & a_{nn} \end{bmatrix}~.

Note that the diagonal elements of a skew-symmetric matrix have to be zero: a_{ii} = -a_{ii} \Rightarrow a_{ii} = 0.

Matrix addition

Let \mathbf{A} and \mathbf{B} be two m \times n matrices with components aij and bij, respectively. Then

 \mathbf{C} = \mathbf{A} + \mathbf{B} \implies c_{ij} = a_{ij} + b_{ij}

Multiplication by a scalar

Let \mathbf{A} be a m \times n matrix with components aij and let λ be a scalar quantity. Then,

 \mathbf{C} = \lambda\mathbf{A} \implies c_{ij} = \lambda a_{ij}

Multiplication of matrices

Let \mathbf{A} be a m \times n matrix with components aij. Let \mathbf{B} be a p \times q matrix with components bij.

The product \mathbf{C} = \mathbf{A} \mathbf{B} is defined only if n = p. The matrix \mathbf{C} is a m \times q matrix with components cij. Thus,

 \mathbf{C} = \mathbf{A} \mathbf{B} \implies c_{ij} = \sum^n_{k=1} a_{ik} b_{kj}

Similarly, the product \mathbf{D} = \mathbf{B} \mathbf{A} is defined only if q = m. The matrix \mathbf{D} is a p \times n matrix with components dij. We have

 \mathbf{D} = \mathbf{B} \mathbf{A} \implies d_{ij} = \sum^m_{k=1} b_{ik} a_{kj}

Clearly, \mathbf{C} \ne \mathbf{D} in general, i.e., the matrix product is not commutative.

However, matrix multiplication is distributive. That means

 \mathbf{A} (\mathbf{B} + \mathbf{C}) = \mathbf{A} \mathbf{B} + \mathbf{A} \mathbf{C} ~.

The product is also associative. That means

 \mathbf{A} (\mathbf{B} \mathbf{C}) = (\mathbf{A} \mathbf{B}) \mathbf{C} ~.

Transpose of a matrix

Let \mathbf{A} be a m \times n matrix with components aij. Then the transpose of the matrix is defined as the n \times m matrix \mathbf{B} = \mathbf{A}^T with components bij = aji. That is,

 \mathbf{B} = \mathbf{A}^T = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \dots & a_{1n} \ a_{21} & a_{22} & a_{23} & \dots & a_{2n} \ a_{31} & a_{32} & a_{33} & \dots & a_{3n} \ \vdots & \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & a_{m3} & \dots & a_{mn} \end{bmatrix}^T = \begin{bmatrix} a_{11} & a_{21} & a_{31} & \dots & a_{m1} \ a_{12} & a_{22} & a_{32} & \dots & a_{m2} \ a_{13} & a_{23} & a_{33} & \dots & a_{m3} \ \vdots & \vdots & \vdots & \ddots & \vdots \ a_{1n} & a_{2n} & a_{3n} & \dots & a_{mn} \end{bmatrix}

A important identity involving the transpose of matrices is

 { (\mathbf{A} \mathbf{B})^T = \mathbf{B}^T \mathbf{A}^T }~.

Determinant of a matrix

The determinant of a matrix is defined only for square matrices.

For a 2 \times 2 matrix \mathbf{A}, we have

 \mathbf{A} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \implies \det(\mathbf{A}) = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{vmatrix} = a_{11} a_{22} - a_{12} a_{21} ~.

For a n \times n matrix, the determinant is calculated by expanding into minors as

\begin{align} &\det(\mathbf{A}) = \begin{vmatrix} a_{11} & a_{12} & a_{13} & \dots & a_{1n} \ a_{21} & a_{22} & a_{23} & \dots & a_{2n} \ a_{31} & a_{32} & a_{33} & \dots & a_{3n} \ \vdots & \vdots & \vdots & \ddots & \vdots \ a_{n1} & a_{n2} & a_{n3} & \dots & a_{nn} \end{vmatrix} \ &= a_{11} \begin{vmatrix} a_{22} & a_{23} & \dots & a_{2n} \ a_{32} & a_{33} & \dots & a_{3n} \ \vdots & \vdots & \ddots & \vdots \ a_{n2} & a_{n3} & \dots & a_{nn} \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} & \dots & a_{2n} \ a_{31} & a_{33} & \dots & a_{3n} \ \vdots & \vdots & \ddots & \vdots \ a_{n1} & a_{n3} & \dots & a_{nn} \end{vmatrix} + \dots \pm a_{1n} \begin{vmatrix} a_{21} & a_{23} & \dots & a_{2(n-1)} \ a_{31} & a_{33} & \dots & a_{3(n-1)} \ \vdots & \vdots & \ddots & \vdots \ a_{n1} & a_{n3} & \dots & a_{n(n-1)} \end{vmatrix} \end{align}

In short, the determinant of a matrix \mathbf{A} has the value

 { \det(\mathbf{A}) = \sum^n_{i=1} (-1)^{i+j} a_{ij} M_{ij} }

where Mij is the determinant of the submatrix of \mathbf{A} formed by eliminating row i and column j from \mathbf{A}.

Some useful identities involving the determinant are given below.

  • If \mathbf{A} is a n \times n matrix, then
 \det(\mathbf{A}) = \det(\mathbf{A}^T)~.
  • If λ is a constant and \mathbf{A} is a n \times n matrix, then
 \det(\lambda\mathbf{A}) = \lambda^n\det(\mathbf{A}) \implies \det(-\mathbf{A}) = (-1)^n\det(\mathbf{A}) ~.
  • If \mathbf{A} and \mathbf{B} are two n \times n matrices, then
 \det(\mathbf{A}\mathbf{B}) = \det(\mathbf{A})\det(\mathbf{B})~.

Inverse of a matrix

Let \mathbf{A} be a n \times n matrix. The inverse of \mathbf{A} is denoted by \mathbf{A}^{-1} and is defined such that

 { \mathbf{A} \mathbf{A}^{-1} = \mathbf{I} }

where \mathbf{I} is the n \times n identity matrix.

The inverse exists only if \det(\mathbf{A}) \ne 0. A singular matrix does not have an inverse.

An important identity involving the inverse is

 { (\mathbf{A}\mathbf{B})^{-1} = \mathbf{B}^{-1} \mathbf{A}^{-1}, }

since this leads to:  { (\mathbf{A} \mathbf{B})^{-1} (\mathbf{A} \mathbf{B}) = (\mathbf{B}^{-1} \mathbf{A}^{-1}) (\mathbf{A} \mathbf{B} ) = \mathbf{B}^{-1} \mathbf{A}^{-1} \mathbf{A} \mathbf{B} = \mathbf{B}^{-1} (\mathbf{A}^{-1} \mathbf{A}) \mathbf{B} = \mathbf{B}^{-1} \mathbf{I} \mathbf{B} = \mathbf{B}^{-1} \mathbf{B} = \mathbf{I}. }

Some other identities involving the inverse of a matrix are given below.

  • The determinant of a matrix is equal to the multiplicative inverse of the

determinant of its inverse.

 \det(\mathbf{A}) = \cfrac{1}{\det(\mathbf{A}^{-1})}~.
  • The determinant of a similarity transformation of a matrix

is equal to the original matrix.

 \det(\mathbf{B} \mathbf{A} \mathbf{B}^{-1}) = \det(\mathbf{A}) ~.

We usually use numerical methods such as Gaussian elimination to compute the inverse of a matrix.

Eigenvalues and eigenvectors

A thorough explanation of this material can be found at Eigenvalue, eigenvector and eigenspace. However, for further study, let us consider the following examples:

  • Let : \mathbf{A} = \begin{bmatrix} 1 & 6 \\ 5 & 2 \end{bmatrix} , \mathbf{v} = \begin{bmatrix} 6 \\ -5 \end{bmatrix} , \mathbf{t} = \begin{bmatrix} 7 \\ 4 \end{bmatrix}~.

Which vector is an eigenvector for  \mathbf{A}  ?

We have  \mathbf{A}\mathbf{v} = \begin{bmatrix} 1 & 6 \\ 5 & 2 \end{bmatrix}\begin{bmatrix} 6 \\ -5 \end{bmatrix} = \begin{bmatrix} -24 \\ 20 \end{bmatrix} = 4\begin{bmatrix} 6 \\ -5 \end{bmatrix} , and  \mathbf{A}\mathbf{t} = \begin{bmatrix} 1 & 6 \\ 5 & 2 \end{bmatrix}\begin{bmatrix} 7 \\ 4 \end{bmatrix} = \begin{bmatrix} 31 \\ 43 \end{bmatrix}~.

Thus,  \mathbf{v} is an eigenvector.

  • Is  \mathbf{u} = \begin{bmatrix} 1 \\ 4 \end{bmatrix} an eigenvector for  \mathbf{A} = \begin{bmatrix} -3 & -3 \\ 1 & 8 \end{bmatrix} ?

We have that since  \mathbf{A}\mathbf{u} = \begin{bmatrix} -3 & -3 \\ 1 & 8 \end{bmatrix}\begin{bmatrix} 1 \\ 4 \end{bmatrix} = \begin{bmatrix} -15 \\ 33 \end{bmatrix} ,  \mathbf{u} = \begin{bmatrix} 1 \\ 4 \end{bmatrix} is not an eigenvector for  \mathbf{A} = \begin{bmatrix} -3 & -3 \\ 1 & 8 \end{bmatrix}~.

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