REVIEW O F MATRIX THEORY in VS .NET

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A.2 TRANSPOSE AND INVERSE A. Transpose:

Let A be an n x m matrix. The transpose of A, denoted by AT, is an m x n matrix formed by interchanging the rows and columns of A.

1) 11

( A .14)

Note that if a matrix is skew-symmetric, then its diagonal elements are all zero. Notes:

( A .1.5)

B. Inverses:

A matrix A is said to be invertible if there exists a matrix B such that BA=AB=I The matrix B is called the inverse of A and is denoted by A-I. Thus,

( A .16a)

( A .l6b)

Thus,

Notes:

1. ( A - y = A 2. ( A - ' ) ~ = ( A ~ ) - '

LINEAR INDEPENDENCE AND RANK Linear independence:

Let A = [ a , a , ... a,], where ai denotes the ith column vector of A. A set of column vectors a; ( 1 = 1 , 2 , . . . ,n) is said to be linearly dependent if there exist numbers a i (i = 1,2,. . .,n ) not all zero such that

= 0,

Since 2a,

+ (-3)a, + a, = 0, a,, a,,

B. Rank of a Matrix:

The number of linearly independent column vectors in a matrix A is called the column rank of A, and the number of linearly independent row vectors in a matrix A is called the row rank of A. It can be shown that Rank of A = column rank of A = row rank of A

Note:

(A.19)

DETERMINANTS Definitions:

Let A = [aij] be a square matrix of order N. We associate with A a certain number called its determinant, denoted by detA or IAl. Let M,, be the square matrix of order ( N - 1) obtained from A by deleting the ith row and jth column. The number A,j defined by

j = 1 9 2 . .

(A.21~)

C akjAk,

( A.21b)

Equation ( A . 2 1 ~ ) known as the Laplace expansion of IAl along the ith row, and Eq. is (A.21b) the Laplace expansion of IAl along the jth column. EXAMPLE A.10 For a 1 x 1 matrix,

A = [ a , , ] --, IAl = a , ,

For a 2 x 2 matrix,

For a 3 x 3 matrix,

B. Determinant Rank of a Matrix:

The determinant rank of a matrix A is defined as the order of the largest square submatrix M of A such that det M # 0. It can be shown that the rank of A is equal to the determinant rank of A.

Hence the rank of the matrix A is 2. (See Example A.9.)

C. Inverse of a Matrix:

a d j A = [ ~ , , ] ~ = [I ~ : A A N :

... ...

where A,, is the cofactor of a i j defined in Eq. (A.20) and "adj" stands for the adjugate (or adjoint). Formula (A.25) is used mainly for N = 2 and N = 3.

APP. A]

Thus,

For a 2 x 2 matrix,

From Eq. (A.25)we see that if det A = 0, then A-' does not exist. The matrix A is called singular if det A = 0, and nonsingular if det A # 0. Thus, if a matrix is nonsingular, then it is invertible and A-' exists.

EIGENVALUES AND EIGENVECTORS Definitions:

(A.28)

for some scalar A and nonzero column vector x, then A is called an eigenvalue (or characteristic value) of A and x is called an eigenuector associated with A. B.

Characteristic Equation:

Equation (A.28) can be rewritten as (A1 - A ) x = 0 (A.29) where I is the identity matrix of Nth order. Equation (A.29) will have a nonzero 1 eigenvector x only if A - A is singular, that is,