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Appendix F

Review of Matrix Algebra

In this section we present some basic definitions for matrix algebra.

Matrix

A matrix is a collection of real or complex numbers arranged to form a rectangular grid:

\begin{matrix} A = [\begin{array}{l} a_{11} & a_{12} & ... & a_{1 M} \\ a_{21} & a_{22} & ... & a_{2 M} \\ ⋮ \\ a_{N 1} & a_{N 2} & ... & a_{N M} \end{array}] & (F .1) \end{matrix}

$\begin{matrix} A = [\begin{array}{l} a_{11} & a_{12} & ... & a_{1 M} \\ a_{21} & a_{22} & ... & a_{2 M} \\ ⋮ \\ a_{N 1} & a_{N 2} & ... & a_{N M} \end{array}] & (F .1) \end{matrix}$

A total of NM numbers arranged into N rows and M columns as shown in Eqn. (F.1) form a N × M matrix. The numbers that make up the matrix are referred to as the elements of the matrix. Two subscripts are used to specify the placement of an element in the matrix. The element at the in row-i and column-j is denoted by aij.

Vector

A vector is a special matrix with only one row or only one column. A matrix with only one row (a 1 × N matrix) is called a row vector of length N. A matrix with only one column (a M × 1 matrix) is called a column vector of length M.

Square matrix

A square matrix is one in which the number of rows is equal to the number of columns. An example is

\begin{matrix} A = [\begin{array}{l} 5 & 2 & 0 & 4 \\ 3 & - 1 & 0 & 0 \\ 0 & 1 & 1 & 3 \\ - 7 & 4 & 3 & 5 \end{array}] & (F .2) \end{matrix}

$\begin{matrix} A = [\begin{array}{l} 5 & 2 & 0 & 4 \\ 3 & - 1 & 0 & 0 \\ 0 & 1 & 1 & 3 \\ - 7 & 4 & 3 & 5 \end{array}] & (F .2) \end{matrix}$

Diagonal matrix

A diagonal matrix is a square matrix in which all elements that are not on the main diagonal are equal to zero. The main diagonal of a square matrix is the diagonal from the top left corner to the bottom right corner. Elements on the main diagonal have identical row and column indices such as a11, a22,...,aN N. An example is

\begin{matrix} A = [\begin{array}{l} 5 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 4 \end{array}] & (F .3) \end{matrix}

$\begin{matrix} A = [\begin{array}{l} 5 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 4 \end{array}] & (F .3) \end{matrix}$

Identity matrix

An identity matrix is a diagonal matrix in which all diagonal elements are equal to unity. For example, the identity matrix of order 3 is

\begin{matrix} I_{3} = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] & (F .4) \end{matrix}

$\begin{matrix} I_{3} = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] & (F .4) \end{matrix}$

Equality of two matrices

Two matrices A and B are equal to each other if they have the same dimensions and if all corresponding elements are equal, that is

\begin{matrix} A = B & \Rightarrow & a_{i j} = b_{i j} & \begin{matrix} for & i = 1, ..., M & \begin{matrix} and & j = 1, ..., N & \begin{matrix} (F .5) \end{matrix} \end{matrix} \end{matrix} \end{matrix}

$\begin{matrix} A = B & \Rightarrow & a_{i j} = b_{i j} & \begin{matrix} for & i = 1, ..., M & \begin{matrix} and & j = 1, ..., N & \begin{matrix} (F .5) \end{matrix} \end{matrix} \end{matrix} \end{matrix}$

Trace of a square matrix

The trace of a square matrix is defined as the arithmetic sum of all of its elements on the main diagonal.

\begin{matrix} Trace (A) = \sum_{i = 1}^{N} a_{i i} & (F .6) \end{matrix}

$\begin{matrix} Trace (A) = \sum_{i = 1}^{N} a_{i i} & (F .6) \end{matrix}$

Transpose of a matrix

The transpose of a matrix A is denoted by AT, and is obtained by interchanging rows and columns so that row-i of A becomes column-i of AT. For example, the transpose of the matrix A in Eqn. (F.2) is

\begin{matrix} A^{T} = [\begin{array}{l} 5 & 3 & 0 & - 7 \\ 2 & - 1 & 1 & 4 \\ 0 & 0 & 1 & 3 \\ 4 & 0 & 3 & 5 \end{array}] & (F .7) \end{matrix}

$\begin{matrix} A^{T} = [\begin{array}{l} 5 & 3 & 0 & - 7 \\ 2 & - 1 & 1 & 4 \\ 0 & 0 & 1 & 3 \\ 4 & 0 & 3 & 5 \end{array}] & (F .7) \end{matrix}$

Determinant of a square matrix

Each square matrix has a scalar value associated with it referred to as the determinant. Methods for computing determinants can be found in most texts on linear algebra. As an example, the determinant of the matrix

\begin{matrix} A = [\begin{array}{l} 5 & 2 \\ 3 & - 1 \end{array}] & (F .8) \end{matrix}

$\begin{matrix} A = [\begin{array}{l} 5 & 2 \\ 3 & - 1 \end{array}] & (F .8) \end{matrix}$

is computed as

\begin{matrix} | A | = (5) (- 1) - (2) (3) = - 1 & (F .9) \end{matrix}

$\begin{matrix} | A | = (5) (- 1) - (2) (3) = - 1 & (F .9) \end{matrix}$

Minors of a square square matrix

The row-i column- j minor mij of a square matrix A is defined as the determinant of the submatrix obtained by deleting row-i and column-j from the matrix A. Consider, for example the square matrix

\begin{matrix} A = [\begin{array}{l} 5 & 2 & 4 \\ 3 & - 1 & 2 \\ - 1 & 0 & 3 \end{array}] & (F .10) \end{matrix}

$\begin{matrix} A = [\begin{array}{l} 5 & 2 & 4 \\ 3 & - 1 & 2 \\ - 1 & 0 & 3 \end{array}] & (F .10) \end{matrix}$

The minor m23 is found by deleting row-2 and column-3 and computing the determinant of the matrix left behind as

\begin{matrix} m_{23} = | \begin{array}{l} 5 & 2 \\ - 1 & 0 \end{array} | = 2 & (F .11) \end{matrix}

$\begin{matrix} m_{23} = | \begin{array}{l} 5 & 2 \\ - 1 & 0 \end{array} | = 2 & (F .11) \end{matrix}$

Cofactors of a square square matrix

The row- i column-j cofactor Δij of a square matrix A is defined as

\begin{matrix} Δ_{i j} = {(- 1)}^{i + j} m_{i j} & (F .12) \end{matrix}

$\begin{matrix} Δ_{i j} = {(- 1)}^{i + j} m_{i j} & (F .12) \end{matrix}$

Using the matrix in Eqn. (F.10) as an example, the cofactor Δ23 is

\begin{matrix} Δ_{23} = {(- 1)}^{2 + 3} m_{23} = - 2 & (F .13) \end{matrix}

$\begin{matrix} Δ_{23} = {(- 1)}^{2 + 3} m_{23} = - 2 & (F .13) \end{matrix}$

Adjoint of a square square matrix

The adjoint of a square matrix A is found by first transposing A and then replacing each element by its cofactor. For example, the adjoint of the matrix A in Eqn. (F.10) is

\begin{matrix} Adj (A) = [\begin{array}{l} - 3 & - 6 & 8 \\ - 11 & 19 & 2 \\ - 1 & - 2 & - 11 \end{array}] & (F .14) \end{matrix}

$\begin{matrix} Adj (A) = [\begin{array}{l} - 3 & - 6 & 8 \\ - 11 & 19 & 2 \\ - 1 & - 2 & - 11 \end{array}] & (F .14) \end{matrix}$

Scaling a matrix

Scaling a matrix by a constant scale factor means multiplying each element of the matrix with that scale factor.

Addition of two matrices

Addition of two matrices is only defined for matrices with identical dimensions. The sum of two matrices A and B is the matrix C in which each element is equal to the sum of corresponding elements of A and B.

\begin{matrix} C = A + B & \Rightarrow & c_{i j} = a a_{i j} + b_{i j} & \begin{matrix} for & i = 1, ..., M & \begin{matrix} and & j = 1, ..., N & \begin{matrix} (F .15) \end{matrix} \end{matrix} \end{matrix} \end{matrix}

$\begin{matrix} C = A + B & \Rightarrow & c_{i j} = a a_{i j} + b_{i j} & \begin{matrix} for & i = 1, ..., M & \begin{matrix} and & j = 1, ..., N & \begin{matrix} (F .15) \end{matrix} \end{matrix} \end{matrix} \end{matrix}$

Scalar product of two vectors

The scalar product of a 1 × N row vector x and a N × 1 column vector y is defined as

\begin{matrix} x \cdot y = \sum_{i = 1}^{N} x_{i} y_{i} & (F .16) \end{matrix}

$\begin{matrix} x \cdot y = \sum_{i = 1}^{N} x_{i} y_{i} & (F .16) \end{matrix}$

Multiplication of two matrices

The product of an M × K matrix A and a K × N matrix B is a M × N matrix C the elements of which are computed as

\begin{matrix} c_{i j} = \sum_{k = 1}^{k} a_{i k} b_{k j} & (F .17) \end{matrix}

$\begin{matrix} c_{i j} = \sum_{k = 1}^{k} a_{i k} b_{k j} & (F .17) \end{matrix}$

In other words, the row-i column-j element of matrix C is the scalar product of row-i of of matrix A and column-j of matrix B.

Inversion of a matrix

The inverse of a square matrix A denoted by A−1 is also a square matrix. It that satisfies the equation

\begin{matrix} A^{- 1} A = {AA}^{- 1} = I & (F .18) \end{matrix}

$\begin{matrix} A^{- 1} A = {AA}^{- 1} = I & (F .18) \end{matrix}$

It is computed by scaling the adjoint matrix by the reciprocal of the determinant, that is,

\begin{matrix} A^{- 1} = \frac{Adj (A)}{| A |} & (F .19) \end{matrix}

$\begin{matrix} A^{- 1} = \frac{Adj (A)}{| A |} & (F .19) \end{matrix}$

The inverse does not exist if the determinant of A is equal to zero. Such a matrix is called a singular matrix.

Eigenvalues and eigenvectors

The scalar λ and the N × 1 vector v are called an eigenvalue and an eigenvector of the N × N matrix A respectively if they satisfy the equation

\begin{matrix} Av = λ v & (F .20) \end{matrix}

$\begin{matrix} Av = λ v & (F .20) \end{matrix}$

An N × N matrix A has N eigenvalues and associated eigenvectors although they are not necessarily unique.

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Table of Contents for Appendix F Review of Matrix Algebra

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