Introduction to Data Compression, 5th Edition

Appendix B

A Brief Review of Matrix Concepts

In this appendix we will look at some of the basic concepts of matrix algebra. Our intent is simply to familiarize you with some basic matrix operations that we will need in our study of compression. Matrices are very useful for representing linear systems of equations, and matrix theory is a powerful tool for the study of linear operators. In our study of compression techniques we will use matrices both in the solution of systems of equations and in our study of linear transforms.

B.1 A Matrix

A collection of real or complex elements arranged in M rows and N columns is called a matrix of order $M\times N$

(B.1) $\mathbf{A}=\left[\begin{array}{cccc}\hfill a_{00}\hfill & \hfill a_{01}\hfill & \hfill \cdots \hfill & \hfill a_{0N-1}\hfill \\ \hfill a_{10}\hfill & \hfill a_{11}\hfill & \hfill \cdots \hfill & \hfill a_{1N-1}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill a_{(M-1)0}\hfill & \hfill a_{(M-1)1}\hfill & \hfill \cdots \hfill & \hfill a_{M-1N-1}\hfill \end{array}\right]$

(B.1)

where the first subscript ...