Page 43 - Jolliffe I. Principal Component Analysis

2. Properties of Population Principal Components
12
Proof. Let β be the kth column of B; as the columns of A form a basis
k
for p-dimensional space, we have
p

β = c jk α j , k =1, 2,... ,q,
k
j=1
where c jk ,j =1, 2,... ,p, k =1, 2,... ,q, are appropriately defined con-
stants. Thus B = AC, where C is the (p × q) matrix with (j, k)th element
c jk ,and
B ΣB = C A ΣAC = C ΛC, using (2.1.3)

p

= λ j c j c
j
j=1
where c is the jth row of C. Therefore

j
p

tr(B ΣB)= λ j tr(c j c )

j
j=1
p

= λ j tr(c c j )

j
j=1
p

=
λ j c c j
j
j=1
p q
2
= λ j c . (2.1.6)
jk
j=1 k=1
Now

C = A B, so

C C = B AA B = B B = I q ,

because A is orthogonal, and the columns of B are orthonormal. Hence
p q
2
c = q, (2.1.7)
jk
j=1 k=1
and the columns of C are also orthonormal. The matrix C can be thought
of as the first q columns of a (p × p) orthogonal matrix, D,say.But the

rows of D are orthonormal and so satisfy d d j =1,j =1,...,p.Asthe
j
rows of C consist of the first q elements of the rows of D, it follows that

c c j ≤ 1,j =1,...,p, that is
j
q
2
c ≤ 1. (2.1.8)
jk
k=1