Page 55 - Introduction to Statistical Pattern Recognition

2 Random Vectors and their Properties 37

Theorem If A and 0 are the eigenvalue and eigenvector matrices of Q, the
eigenvalue and eigenvector matrices of Q"' for any integer m are A"' and @ respec-
tively. That is,

Q@=@A + Qm@ =@Am. (2.129)

Proof Using QQ =@A,

Theorem The trace of Q" is the summation of hy's, and invariant under
any orthonormal transformation. That is,
I1
trQ" = tr A"' = xhr . (2. I3 I)
i=l

Example 6: Let us consider n eigenvalues, h I , . . . , h,, , as the samples drawn
from the distribution of a random variable h. Then we can calculate all sample
moments of the distribution of 1 by

1
1 I1
E(P} =-XI:' = -trQn', (2.132)
n ;=I n
where E {. ) indicates the sample estimate of E { }. Particularly, we may use
A 1 1
E(1J -trQ = -Cqji, (2.133)
=
n n j=l
1 1
car(X} = -trQ2 - (-trQ j2
n n

(2.134)

Example 7: Equation (2.13 1 ) is used to find the largest eigenvalue because

A;' + . , . + hr E h;' for m >>1 , (2.135)
where h, is assumed to be the largest eigenvalue. For example, if we select