[B.6-1 Some Definitions and Properties](#page-6-0)

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B.6-1 Some Definitions and Properties

A square matrix whose elements are zero everywhere except on the main diagonal is a diagonal matrix. An example of a diagonal matrix is

⎡ 2	0	0	⎤
0 ⎣	1	0	⎦
0	0	5

A diagonal matrix with unity for all its diagonal elements is called an identity matrix or a unit matrix, denoted by I. This is a square matrix:

	⎡ 1	0	0	···	⎤ 0
	0 ⎢	1	0	···	0 ⎥
I =	⎢ 0 ⎢	0	1	···	⎥ 0 ⎥
	⎢ ⎢ ⎣			···	⎥ ⎥ ⎦
	0	0	0	···	1

The order of the unit matrix is sometimes indicated by a subscript. Thus, I*ⁿ* represents the n×n unit matrix (or identity matrix). However, we shall omit the subscript since order is easily understood by context.

A matrix having all its elements zero is a zero matrix.

A square matrix A is a symmetric matrix if aij = aji (symmetry about the main diagonal).

Two matrices of the same order are said to be equal if they are equal element by element. Thus, if

$$\mathbf{A} = (a_{ij})_{m \times n} \quad \text{and} \quad \mathbf{B} = (b_{ij})_{m \times n}$$

then A = B only if aij = bij for all i and j.

If the rows and columns of an m×n matrix A are interchanged so that the elements in the ith row now become the elements of the ith column (for i = 1, 2,…,m), the resulting matrix is called the transpose of ^A and is denoted by ^AT . It is evident that ^AT is an ⁿ×^m matrix. For example, if

$$\mathbf{A} = \begin{bmatrix} 2 & 1 \\ 3 & 2 \\ 1 & 3 \end{bmatrix}, \quad \text{then} \quad \mathbf{A}^T = \begin{bmatrix} 2 & 3 & 1 \\ 1 & 2 & 3 \end{bmatrix}$$

38 CHAPTER B BACKGROUND

Using the abbreviated notation, if ^{A =} (aij)^m×ⁿ, then ^{AT =} (aji)ⁿ×^m. Intuitively, further notice that (A*^T* )^{T = A}.

B.6-2 Matrix Algebra

We shall now define matrix operations, such as addition, subtraction, multiplication, and division of matrices. The definitions should be formulated so that they are useful in the manipulation of matrices.

ADDITION OF MATRICES

For two matrices A and B, both of the same order (m×n),

$$\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \text{ and } \mathbf{B} = \begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1n} \\ b_{21} & b_{22} & \cdots & b_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ b_{m1} & b_{m2} & \cdots & b_{mn} \end{bmatrix}$$

we define the sum A+B as

$$\mathbf{A} + \mathbf{B} = \begin{bmatrix} (a_{11} + b_{11}) & (a_{12} + b_{12}) & \cdots & (a_{1n} + b_{1n}) \\ (a_{21} + b_{21}) & (a_{22} + b_{22}) & \cdots & (a_{2n} + b_{2n}) \\ \vdots & \vdots & \ddots & \vdots \\ (a_{m1} + b_{m1}) & (a_{m2} + b_{m2}) & \cdots & (a_{mn} + b_{mn}) \end{bmatrix}$$

or

$$\mathbf{A} + \mathbf{B} = (a_{ij} + b_{ij})_{m \times n}$$

Note that two matrices can be added only if they are of the same order.

MULTIPLICATION OF A MATRIX BY A SCALAR

We multiply a matrix A by a scalar c as follows:

$$c\mathbf{A} = c \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} = \begin{bmatrix} ca_{11} & ca_{12} & \cdots & ca_{1n} \\ ca_{21} & ca_{22} & \cdots & ca_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ ca_{m1} & ca_{m2} & \cdots & ca_{mn} \end{bmatrix} = \mathbf{A}c$$

Thus, we also observe that the scalar c and the matrix A commute: cA = Ac.

MATRIX MULTIPLICATION

We define the product

$$AB = C$$

in which cij, the element of C in the ith row and jth column, is found by adding the products of the elements of A in the ith row multiplied by the corresponding elements of B in the jth column. Thus,

$$c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \dots + a_{in}b_{nj} = \sum_{k=1}^{n} a_{ik}b_{kj}$$

(B.33)

This result is expressed as follows:

Note carefully that if this procedure is to work, the number of columns of A must be equal to the number of rows of B. In other words, AB, the product of matrices A and B, is defined only if the number of columns of A is equal to the number of rows of B. If this condition is not satisfied, the product AB is not defined and is meaningless. When the number of columns of A is equal to the number of rows of B, matrix A is said to be conformable to matrix B for the product AB. Observe that if A is an m × n matrix and B is an n × p matrix, A and B are conformable for the product, and C is an m×p matrix.

We demonstrate the use of the rule in Eq. (B.33) with the following examples.

$$\begin{bmatrix} 2 & 3 \\ 1 & 1 \\ 3 & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 1 & 2 \\ 2 & 1 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 8 & 9 & 5 & 7 \\ 3 & 4 & 2 & 3 \\ 5 & 10 & 4 & 7 \end{bmatrix}$$ $$\begin{bmatrix} 2 & 1 & 3 \end{bmatrix} \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} = 8$$

In both cases, the two matrices are conformable. However, if we interchange the order of the first matrices as follows:

1	3	1	!⎡ 2	2	3	⎤
				1 ⎣	1	⎦
2	1	1	1	3	1

the matrices are no longer conformable for the product. It is evident that, in general,

AB = BA

Indeed, AB may exist and BA may not exist, or vice versa, as in our examples. We shall see later that for some special matrices, AB = BA. When this is true, matrices A and B are said to commute. We re-emphasize that in general, matrices do not commute.

40 CHAPTER B BACKGROUND

In the matrix product AB, matrix A is said to be postmultiplied by B or matrix B is said to be premultiplied by A. We may also verify the following relationships:

$$(A + B)C = AC + BC$$ $$C(A + B) = CA + CB$$

We can verify that any matrix A premultiplied or postmultiplied by the identity matrix I remains unchanged:

AI = IA = A

Of course, we must make sure that the order of I is such that the matrices are conformable for the corresponding product.

We give here, without proof, another important property of matrices:

$$|\mathbf{A}\mathbf{B}| = |\mathbf{A}||\mathbf{B}|$$

where |A| and |B| represent determinants of matrices A and B.

MULTIPLICATION OF A MATRIX BY A VECTOR

Consider Eq. (B.32), which represents Eq. (B.31). The right-hand side of Eq. (B.32) is a product of the m×n matrix A and a vector x. If, for the time being, we treat the vector x as if it were an n×1 matrix, then the product Ax, according to the matrix multiplication rule, yields the right-hand side of Eq. (B.31). Thus, we may multiply a matrix by a vector by treating the vector as if it were an n × 1 matrix. Note that the constraint of conformability still applies. Thus, in this case, xA is not defined and is meaningless.

MATRIX INVERSION

To define the inverse of a matrix, let us consider the set of equations represented by Eq. (B.32) when m = n:

$$\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$$

(B.34)

We can solve this set of equations for x1, x2, … , xn in terms of y1, y2, … , yn by using Cramer’s rule [see Eq. (B.21)]. This yields

$$\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} \frac{|\mathbf{D}_{11}|}{|\mathbf{A}|} & \frac{|\mathbf{D}_{21}|}{|\mathbf{A}|} & \cdots & \frac{|\mathbf{D}_{n1}|}{|\mathbf{A}|} \\ \frac{|\mathbf{D}_{12}|}{|\mathbf{A}|} & \frac{|\mathbf{D}_{22}|}{|\mathbf{A}|} & \cdots & \frac{|\mathbf{D}_{n2}|}{|\mathbf{A}|} \\ \vdots & \vdots & \cdots & \vdots \\ \frac{|\mathbf{D}_{1n}|}{|\mathbf{A}|} & \frac{|\mathbf{D}_{2n}|}{|\mathbf{A}|} & \cdots & \frac{|\mathbf{D}_{nn}|}{|\mathbf{A}|} \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}$$

(B.35)

in which |A| is the determinant of the matrix A and |Dij| is the cofactor of element aij in the matrix A. The cofactor of element aij is given by (−1)ⁱ+^j times the determinant of the (n − 1) × (n − 1) matrix that is obtained when the ith row and the jth column in matrix A are deleted.

We can express Eq. (B.34) in compact matrix form as

$$y = Ax$$

(B.36)

We now define A−¹ , the inverse of a square matrix A, with the property

A−¹ A = I (unit matrix)

Then, premultiplying both sides of Eq. (B.36) by A−¹ , we obtain

$$\mathbf{A}^{-1}\mathbf{y} = \mathbf{A}^{-1}\mathbf{A}\mathbf{x} = \mathbf{I}\mathbf{x} = \mathbf{x}$$

or

$$\mathbf{x} = \mathbf{A}^{-1} \mathbf{y} \tag{B.37}$$

A comparison of Eq. (B.37) with Eq. (B.35) shows that

$$A^{-1} = \frac{1}{|A|} \begin{bmatrix} |D_{11}| & |D_{21}| & \cdots & |D_{n1}| \\ |D_{12}| & |D_{22}| & \cdots & |D_{n2}| \\ \vdots & \vdots & \cdots & \vdots \\ |D_{1n}| & |D_{2n}| & \cdots & |D_{nn}| \end{bmatrix}$$

One of the conditions necessary for a unique solution of Eq. (B.34) is that the number of equations must equal the number of unknowns. This implies that the matrix A must be a square matrix. In addition, we observe from the solution as given in Eq. (B.35) that if the solution is to exist, |A| = 0.† Therefore, the inverse exists only for a square matrix and only under the condition that the determinant of the matrix be nonzero. A matrix whose determinant is nonzero is a nonsingular matrix. Thus, an inverse exists only for a nonsingular, square matrix. Since A−^{1 A = I =} AA−¹ , we further note that the matrices A and A−¹ commute.‡

The operation of matrix division can be accomplished through matrix inversion.

EXAMPLE B.12 Computing the Inverse of a Matrix

Let us find A−¹ if

A = ⎡ ⎣ 211 123 321 ⎤ ⎦

^† These two conditions imply that the number of equations is equal to the number of unknowns and that all the equations are independent.

^‡ To prove AA−^{1 = I}, notice first that we define ^A−1**^A** ^{= I}. Thus, IA ⁼ AI ^{= A}(A−1A) ⁼ (AA−1)A. Subtracting (AA−1)A, we see that IA−(AA−1)^{A =} 0 or (I−AA−1)^{A =} 0. This requires AA−^{1 = I}.

Here,

$$|\mathbf{D}_{11}| = -4, \t |\mathbf{D}_{12}| = 8, \t |\mathbf{D}_{13}| = -4 |\mathbf{D}_{21}| = 1, \t |\mathbf{D}_{22}| = -1, \t |\mathbf{D}_{23}| = -1 |\mathbf{D}_{31}| = 1, \t |\mathbf{D}_{32}| = -5, \t |\mathbf{D}_{33}| = 3 \text{and } |\mathbf{A}| = -4. \text{ Therefore, } \mathbf{A}^{-1} = -\frac{1}{4} \begin{bmatrix} -4 & 1 & 1 \\ 8 & -1 & -5 \\ -4 & -1 & 3 \end{bmatrix}$$