Adaptive Filtering

Appendix 2

Matrix Analysis

A2.1    DEFINITIONS^*

Let A be an m×n matrix with elements a_ij, i = 1, 2,…, m; j = 1, 2,…, n. A shorthand description of A is

${\left[A\right]}_{ij}=a_{ij}$

(A2.1)

The transpose of A, denoted by A^T, is defined as the n×m matrix with elements a_ji or

${\left[A^T\right]}_{ij}=a_{ij}$

(A2.2)

Example A2.1.1

$A=\left[\begin{array}{cc}1& 2\\ 4& 9\\ 3& 1\end{array}\right];A^T=\left[\begin{array}{ccc}1& 4& 3\\ 2& 9& 1\end{array}\right]$

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A square matrix is a matrix in which m = n. A square matrix is symmetric if A^T = A.

The rank of a matrix is the number of linearly independent rows or columns, whichever is less. The inverse of a square n×n matrix A⁻¹ in which

$A^{-1}A=A{A}^{-1}=I$

(A2.3)

where:

I is the identity matrix

$I=\left[\begin{array}{cccc}1& 0& \cdots & 0\\ 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮\\ 0& 0& \cdots & 1\end{array}\right]$

(A2.4)

A matrix A is singular if its inverse does not exist.

The determinant of a square n×n matrix is denoted ...