Mathematical Foundations - Control of Color Imaging Systems: Analysis and Design - page 159

where

f (li) ¼ Q (l) P (li) þ R (l) ( 3 : 181 )

To get the coef cients of the polynomial R(l), (l) , we set l ¼ l i , i ¼ 1, 2, ... , n in the

above equation:

f (l i ) ¼ Q (l i ) P (l i ) þ R (l i ) ¼ R (l i ) ( 3 : 182 )

This yield a set of n simultaneous linear equations that can be solved for
r 0 , r 1 , ... , r n 1 :

2 3 2 3 2 3

1 n 1

1

1 l 1 l l r 0

r 1
r 2
.
r n 1

f (l 1 )
f (l 2 )
f (l 3 )
.
f (l n )

4 5 4 5 4 5

n 1
2

2

1 l 2 l l

3 n 1

3

1 l 3 l l

¼ ( 3 : 183 )

. . . . .

2
n

n 1

1 l n l l

n

Now let us assume that we have repeated eigenvalues. Without loss of generality, we
assume that the n n matrix A has one eigenvalue of multiplicity m and n m
distinct eigenvalues. That is, the eigenvalues of A are

l 1 , l 1 , ... , l 1 , l m þ 1 , l m þ 2 , ... , l n ( 3 : 184 )

| {z }
m

Since l 1 is an eigenvalue with multiplicity of m, then

l 1 ¼

k Q (l) P (l)
dl

d

0 for k ¼ 1, 2, ... , m ( 3 : 185 )

k

Therefore, we have the following n equations for n unknown r 0 , r 1 , ... , r n 1 :

f (l 1 ) ¼ R (l 1 )
f 0 (l 1 ) ¼ R 0 (l 1 )
f 00 (l 1 ) ¼ R 00 (l 1 )
. .

¼

f m 1 (l 1 ) ¼ R m 1

( 3 : 186 )

(l 1 )
f (l m þ 1 ) ¼ R (l m þ 1 )
f (l m þ 2 ) ¼ R (l m þ 2 )
. ¼ .
f (l n ) ¼ R (l n )