B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

Appendix D: Basic Matrix Properties and Operations 881
If A T = A, then we say that A is a symmetric matrix.
· If A H = A, then we say that A is a Hermitian matrix.
· If A consists of only real entries, then it is both Hermitian and symmetric.
D.2 Matrix Product and Properties
For an m x n matrix A and an n x e matrix B with
(D.3)
the matrix product C = A • B has dimension m x e and equals
c - [
C) , l CJ ,2 au
c2,1 c2,2 a2,e
Cm, 1 Cm,2 Cl,n,[
]
where
n
c · · - L a ·kbk ·
lJ - l, J
k=l
(D.4)
In general AB =fa BA . In fact, the products may not even be well defined. To be able to multiply
A and B, the number of columns of A must equal the number of rows of B.
In particular, the product of a row vector and a column vector is
(D.5a)
= < x. y >
(D.5b)
Therefore. x H x = llx ll 2 .
Two vectors x and y are orthogonal ify H x = x H y = 0.
There are several commonly used properties of matrix products:
A(B + C) =AB + AC
A(BC) = (AB)C
(AB)* =A*B*
(AB/ = B T A T
(AB) H = B H A H
(D.6a)
(D.6b)
(D.6c)
(D.6d)
(D.6e)