Mathematical Foundations - Control of Color Imaging Systems: Analysis and Design - page 149

Example 3.40

Find the p ¼ 1, p ¼ 2, p ¼1 , and Frobenius norm of the following matrix:

42

A ¼

67

S OLUTION
The different matrix norms are computed as

X

m

¼

P P

m m

kAk 1 ¼ max

j

a ij max

j

1 a i1

j j

1 a i2

j j ¼ max

j

½ 10 9 ¼ 10

i¼ i¼

i¼ 1

X P P

m

¼

m m

a 1j a 2j

kAk 1 ¼ max

i

a ij max

i

¼ max

i

½ 613 ¼ 13

j¼ 1 j¼ 1

j¼ 1

s

max

42

p

max

4 6

27

kAk 2 ¼ l (A H A) ¼ l

67

s

max

p

86

52 34

¼ l ¼ : 5037 ¼ 9 : 3007

34 53

0
@

1
A

1
2

X X

2 2

p

16

p

105

2

kAk F ¼ a ij ¼ þ 4 þ 36 þ 49 ¼ ¼ 10 : 247

i ¼ 1 j ¼ 1

3.10.2 P RINCIPAL C OMPONENTS A NALYSIS
Principal component analysis (PCA) is a statistical technique to extract patterns,
similarities, and differences hidden in data. The PCA is extremely useful in case of
data of higher dimensions since illustrative techniques are not applicable in case of
data of dimension greater than three. An important application of PCA is data
dimensionality reduction or data compression. This is achieved by reducing the
data dimension with a small loss of information. Let X be an m-dimensional random
vector, and let X 1 , X 2 ,

, X N be N observations of random vector X. The m m
covariance matrix R of these data is estimated using the sample covariance by

...

X

N
n ¼ 1 ( X n m)( X n m) T

1
N

R ¼ ( 3 : 155 )

m

where is the sample mean vector given by

X

N

1
N

m ¼ X n ( 3 : 156 )

n ¼ 1