Page 60 - Introduction to Statistical Pattern Recognition

42 Introduction to Statistical Pattern Recognition

matrix of Y after the diagonizing transformation. For two distributions, the dis-
tance functions are, by simultaneous diagonalization,

(2.155)

(2.156)

When distance computations are heavily involved in practice, it is suggested to
transform the original data samples Xi to Yj before processing the data. This saves
a significant amount of computation time.

Relation between S-' and C-': We show the inverse matrix of an auto-
correlation matrix in terms of the covariance matrix and expected vector. From
(2.15),
s-I = (C + MM')-' . (2.157)
Applying the simultaneous diagonalization of (2.101) for XI =X and C2 =MM7,
we have A '(C + MMT)A =I + A, or C + MM' = (A ')-I (I + A)A-'. Taking the
inverse,
(Z + MM')-I = A(I + A)-'AT . (2.1 58)

where A is given in (2.141) and (2.142). Therefore,

1
.+A, 0 0
1 +h,
1
1
(I + A)-' =

0 1
0 1