Page 56 - Introduction to Statistical Pattern Recognition

38 Introduction to Statistical Pattern Recognition

m = 16, we need to multiply matrices four times as Q + Q2 -+ Q4 + Q8 + Q 16,
and take the trace of Q l6 to estimate the largest eigenvalue.

Determinant and Rank

Theorem The determinant of Q is equal to the product of all eigenvalues
and is invariant under any orthonormal transformation. That is,
n
lQ I = Ihl = nhi (2.136)
i=l

Proof Since the determinant of the product of matrices is the product of the
determinants of the matrices,
Ihl = IaTI lQl101 = lQl IQTI 101 = lQl . (2.137)

Theorem The rank of Q is equal to the number of nonzero eigenvalues.

Proof Q can be expressed by

(2.138)

where the +;'s are linearly independent vectors with mutually orthonormal rela-
tions. Therefore, if we have (n -r) zero hi's, we can express Q by r linearly
independent vectors, which is the definition of rank r.
Three applications of the above theorems are given as follows:

Relation between I S I and I C I : We show the relation between the deter-
minants of the covariance and autocorrelation matrices [from (2.15)]:
IS1 = IC+MMTI. (2.139)

Applying the simultaneous diagonalization of (2.101) for XI =C and & =MMT,
wehaveAr(C+MMT)A =I +A. Therefore,

fi(l + hi)
I1
IZ + MMTI = i=l = In(l + h;)}lCl , (2.140)
IA I* i=l
where I! I A I = I C I is obtained from (2.101). On the other hand, since the rank of
MM' is one, the hi's should satisfy the following conditions