Adaptive Filtering

2

Matrices

2.1    INTRODUCTION

A generalization of a vector, that is, an ordered array of numbers, determines the magnitude and direction of the vector. A further generalization is that of an n × m matrix $A=\left\{a_{ij}\right\}=\left(a_{ij}\right)$, which is shown below:

$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1m}\\ a_{21}& a_{22}& \cdots & a_{2m}\\ ⋮& ⋮& ⋮\\ a_{n1}& a_{n2}& \cdots & a_{nm}\end{array}\right]$

(2.1)

with a typical element a_ij, where $i=1,2,\dots ,n$ and $j=1,2,\dots ,m$.

2.2    GENERAL TYPES OF MATRICES

We shall present two- and three-element matrices for convenience. However, the properties are identical for any size.

2.2.1    DIAGONAL, IDENTITY, AND SCALAR MATRICES

$\begin{array}{l}\text{Diagonal:}A=\left[\begin{array}{cc}{a}_{\text{11}}& \text{0}\\ \text{0}& {a_{\text{22}}}_{}\end{array}\right]\text{;}\text{Identity:}\mathrm{I=}\left[\begin{array}{cc}\text{1}& \text{0}\\ \text{0}& \text{1}\end{array}\right]\text{;}\\ \text{scalar:}A=\left[\begin{array}{cc}a& 0\\ 0& a\end{array}\right]=a\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=aI\end{array}$

(2.2)

We shall also write diag $\left\{a_{11},a_{22},\dots ,a_{nn}\right\}$ in short.

2.2.2    UPPER AND LOWER TRIANGULAR MATRICES

$\text{Lower ...}$