Image Analysis, Classification and Change Detection in Remote Sensing, 3rd Edition

16 Images, Arrays, and Matrices

1.3.3 Singular matrices

If |A| =

0, then A has no inverse and is said to be a singular matrix. If A is

nonsingular, then the equation

Ax = 0 (1.31)

only has the so-called trivial solution x = 0. To see this, multiply from the

left with A

−1

. Then A

−1

Ax = Ix = x = 0.

If A is singular, Equation (1.31) has at least one nont r ivial solution x 6= 0.

This, again, is eas y to see for a 2 × 2 matrix. Suppose |A| = 0. Writing

Equation (1.31) out fully:

a

11

x

1

+ a

12

x

2

= 0

a

21

x

1

+ a

22

x

2

= 0.

To g et a nontrivial s olution, assume without loss of generality that a

12

6= 0.

Just cho ose x

1

= 1. Then the above two equations imply that

x

2

= −

a

11

a