Mathematical Foundations - Control of Color Imaging Systems: Analysis and Design - page 138

Hence,

a 1 ¼W 1 ¼ 1

1
2 (a 1 W 1 þW 2 )

1
2 (

a 2 ¼ ¼ 1 1 þ 71) ¼ 35

1
3 (

1
3 (35

a 3 ¼ a 2 W 1 þ a 1 W 2 þW 3 ) ¼ 1 þ 1 71 þ 110) ¼ 72

Therefore,

3 2

P( l ) ¼ l þ 8 l þ 9 l 2

3.9.4 M ODAL M ATRIX
Let x 1 , x 2 ,

, x n be the n independent eigenvectors of n n matrix A. The n n
matrix M formed by side-by-side stacking of eigenvectors is called the modal matrix
of A:

...

M ¼ x 1 ½ x 2 x n ( 3 : 124 )

Since the n columns of M are linearly independent, it is full rank and hence

invertible.

Example 3.33

Find the modal matrix of

2
4

3
5

2 : 61 : 3 2 : 5

A ¼ 0 : 85 : 4 5

0 : 81 : 4 1

S OLUTION
Matrix A has three independent eigenvectors (see Example 3.31):

2
4

3
5

2
4

3
5

2
4

3
5

1 1

0

1
3
1

x 1 ¼ 1 : 3333 , x 2 ¼ : 5 , and x 3 ¼

1 : 3333 0 : 5

Therefore, the modal matrix is

2
4

3
5

1 1 1

M ¼ x 1

½ x 2 x n ¼ 1 : 333 0 : 53

1 : 333 0 : 51