Mathematical Foundations - Control of Color Imaging Systems: Analysis and Design - page 137

3.9.3 F INDING C HARACTERISTIC P OLYNOMIAL OF A M ATRIX
Let A be an n n matrix. The characteristic polynomial of matrix A can be found by
using the following recursive algorithm. Let W k ¼ Trace( A k

) , k ¼ 1, 2, ... , n, then

the coef cients of the characteristic equation are [3]

a 1 ¼ W 1
a 2 ¼

1

2 ( a 1 W 1 þ W 2 )

1

a 3 ¼

3 ( a 2 W 1 þ a 1 W 2 þ W 3 )

( 3 : 122 )

.

1
n ( a n 1 W 1 þ a n 2 W 2 þþ a 1 W n 1 þ W n )

a n ¼

and

n n 1 n 2

P (l) ¼ l

þ a 1 l þ a 2 l þþa n 1 l þ a n ( : )

3 123

The above algorithm is known as Bocher is formula [4].

'

Example 3.32

Find the characteristic polynomial of the following 3 3 matrix:

2
4

3
5

242
163
1

A ¼

1 5

S OLUTION
The trace of A, A 2 , and A 3 are

2
4

3
5

2
4

3
5 ¼

2
4

3
5

242
163
1

242
163
1

2
7 75

10 14

A 2 ¼ A A ¼

1 5 1 5 83 4

2
4

3
5

2
4

3
5 ¼

2
4

3
5

2
7 75

10 14 242

163
1

8 126 72

A 3 ¼ A 2

A ¼ 28 245 100

83 4 1 5 43 38 127

W 1 ¼ Trace(A) ¼ 1

Trace(A 2 ) W 2 ¼ ¼ 71

Trace(A 3 )

W 3 ¼ ¼ 110