Page 444 - Schaum's Outline of Theory and Problems of Signals and Systems

APP. A] REVIEW OF MATRIX THEORY

It is important to note that AB = 0 does not necessarily imply A = 0 or B = 0.

EXAMPLE A.5 Let

Then

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.
B=AT+b.,=a..
1) 11 (A. 14)

EXAMPLE A.6

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 Let

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~ (A. 1.5)
=
3. (a~)~= QA~
4. (AB)~=B~A~

B. Inverses:
A matrix A is said to be invertible if there exists a matrix B such that

BA=AB=I (A. 16a)
The matrix B is called the inverse of A and is denoted by A-I. Thus,
A-~A=M-~ =I (A. l6b)