Matrices: a brief overview - Signal Processing and Optimization for Transceiver Systems - page 763

T

verified that P P = I , so the permutation matrix is unitary. Given any N × N

matrix A , consider now the product

T .

B = PAP (B . 32)

The i th diagonal element is
b ii =
k

T ] mi =
k

p i,k a k,m [ P p i,k a k,m p i,m . (B . 33)

m m

All terms in the summation are zero except the terms which have k = n i and
m = n i .Thus

b ii = p i,n i a n i ,n i p i,n i = a n i ,n i .
That is, the diagonal elements of B are permuted versions of the diagonal el-
ements of A . In fact the diagonal elements of B are related to the diagonal
elements of A by the same permutation that relates i to n i .

Exchange matrices .Let P e be a permutation matrix obtained from the iden-
tity matrix I by interchanging two rows. For example, let

⎡ ⎤

10000
00100
01000
00010
00001

⎣ ⎦

P e = . (B . 34)

This is equivalent to interchanging the corresponding columns of I ,andthe

e = P e ). Permutation matrices of the above
form are called exchange matrices . The reason is, if we premultiply a matrix
A with P e it is equivalent to interchanging rows k and m of the matrix A . No
other rows are affected. Similarly if we postmultiply A with P e it is equivalent to
interchanging columns k and m of A . It is readily verified from the properties of
P e that the matrix P e AP e (= P e AP

matrix P e is symmetric (i.e., P

e ) has the k th and m th diagonal elements
of A exchanged, with all other diagonal elements unaffected.

B.6 Positive definite matrices

For any N × N matrix P , the scalar

φ = v † Pv (B . 35)

is said to be a quadratic form ,where v is a column vector. When P is Hermitian,
v † Pv is guaranteed to be real. If the Hermitian matrix P is such that v † Pv > 0
for v

= 0 ,wesaythat P is positive definite. If v † Pv ≥

0 for all v , then P
is positive semidefinite. Negative definiteness and semidefiniteness are similarly
defined. If P is positive definite, we write it as

P > 0 , (B . 36)