REVIEW O F MATRIX THEORY in VS .NET

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REVIEW O F MATRIX THEORY
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It is important to note that AB = 0 does not necessarily imply A = 0 or B = 0.
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A.2 TRANSPOSE AND INVERSE A. Transpose:
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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.
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B=AT+b.,=a..
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If AT= A, then A is said to be symmetric, and if AT = -A, then A is said to be skew-symmetric.
EXAMPLE A.7
Then A is a symmetric matrix and B is a skew-symmetric matrix.
Note that if a matrix is skew-symmetric, 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 A-I. 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,
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