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REVIEW O F MATRIX THEORY in VS .NET
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EXAMPLE A.7
Then A is a symmetric matrix and B is a skewsymmetric matrix.
Note that if a matrix is skewsymmetric, then its diagonal elements are all zero. Notes: 1. (AT)T=A 2. (A + B ) = + B~ ~ 3. ( a ~ ) Q A ~ ~ = 4. ( A B ) ~ = B ~ A ~
( 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 AI. Thus, A~A=M~ I =
( A .16a) ( A .l6b) REVIEW OF MATRIX THEORY
[APP. A
EXAMPLE A.8
Thus, Notes: 1. ( A  y = A 2. ( A  ' ) ~ = ( A ~ )  ' Note that if A is invertible, then AB = 0 implies that B = 0 since
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 If Eq. (A.18) holds only for all cui
EXAMPLE A.9
= 0, then the set is said to be linearly independent.
Since 2a, + (3)a, + a, = 0, a,, a,, and a, are linearly dependent. Let
Then
implies that a ,= a, = a, = 0.Thus, d , , d,, and d, are linearly independent.
REVIEW OF MATRIX THEORY
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) If the rank of an N x N matrix A is N , then A is invertible and A' exists.
A.4 A.
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 is called the cofactor of a,,. Then det A is obtained by
detA=IAl=CaikAik
i = 1 , 2 ,..., N
j = 1 9 2 . . (A.21~) detA = (A1 =
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, REVIEW OF MATRIX THEORY
[APP. A
Using Eqs. (A.21~) (A.231, we obtain and
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. EXAMPLE A . l l
Note that IAl = 0. One of the largest submatrices whose determinant is not equal to zero is
Hence the rank of the matrix A is 2. (See Example A.9.) C. Inverse of a Matrix: Using determinants, the inverse of an N x N matrix A can be computed as
1  adj A det A
A11 and
a d j A = [ ~ , , ] ~ = [I ~ : A A N : ... ... ANI ]: A ANN
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. EXAMPLEA.12 Let
APP. A] REVIEW OF MATRIX THEORY
Then
adj 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. AS A.
EIGENVALUES AND EIGENVECTORS Definitions: Let A be an N x N matrix. If
= Ax
(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,

