Matrices: a brief overview - Signal Processing and Optimization for Transceiver Systems - page 753

Appendix B

Matrices: a brief overview

B.1 Introduction

In this appendix we review basic concepts from matrices which are frequently
used in this topic. More details can be found in several standard texts, e.g.,
Franklin [1968], Horn and Johnson [1985], and Golub and Van Loan [1989]. A
matrix with N rows and M columnsisreferredtoasan N

× M matrix. For

example,

⎡ ⎤ ⎡ ⎤

12+ j
13
24

p 00 p 01

⎣ ⎦ = ⎣ ⎦

P = p 10 p 11

− j p 20 p 21

is a 3 2 matrix. Note that the elements are denoted by double subscripts, as

in p km , sometimes with a comma for clarity as in p k,m . Sometimes we use upper
case to denote the elements, as in P km . In some situations we also use [ P ] km to
denote p km .
The notations P

×

T , P ∗ , and P † indicate the transpose, conjugate, and transpose-
conjugate of a matrix, respectively. For the matrix P given above,

⎡ ⎤

= 112
2+ j

,

P † = 112
2

.

12 j

13
24+ j

−

⎣ ⎦ ,

T

P ∗ = P

34 − j − j 34+ j

Some other notations were mentioned in Sec. 1.6, which the reader may want
to review at this time. When M = N we say that P is a square matrix. A
1

× M matrix is said to be a row vector ,andan N × 1 matrix is said to be a

column vector . A matrix with p km =0for k

= m is called a diagonal matrix.

For example,

⎡ ⎤

10 0
04 0
00

⎣ ⎦

P =

− 2

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