Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 40

Equation (2.31) is a concise matrix way of representing the system of equations
given in (2.30). In this particular case, matrix A is 8 2. The inverse of a matrix is
defined only when A is square and jAj 6¼ 0. A very innovative approach was
adapted by Moore and Penrose for inverting a rectangular matrix.
Take (2.31) and premultiply by A t to obtain
A t Ax ¼ A t y: ð 2 : 32 Þ

The matrix A t A is square and can also be written as 4

ð 0 ð 8

0
5

5 Þþ 8
6

A t A ¼ 6 Þþþ 9 8 ð Þð 9 8 Þ

¼ X

8

x i x t ðx i is ith row vector of AÞ: ð 2 : 33 Þ

i ¼ 1

The column vector A t y can be written as

5 8
6

þ 101

0
5

9
8

A t y ¼ 34

¼ X

8

x i y i :

ð 2 : 34 Þ

i ¼ 1

Under the assumption that A t A is non-singular, the inverse can be obtained
recursively or en block. Then the solution for the system (2.30) is given as

1 A t y ¼ A þ y; x ¼ A t A

½ ð 2 : 35 Þ

, A t ½ 1 A t is known as the Moore-Penrose pseudo-inverse [1, 8]. The
solution is x ¼ A þ y . There are two ways of looking at it. One way is that it
corresponds to the best intersection [9] of all the intersections in Figure 2.17, in a

where A þ

40

30

20

10

0

−10

−20

−8 −7 −6 −5 −4 −3 −2

Figure 2.17 A system of linear equations

Þ 1

4 This is useful in inverting recursively using a matrix inversion lemma which states that ðB þ xx t

¼

.

B 1 xx t B 1
1 þx t B 1 x

B 1