Page 59 - Introduction to Statistical Pattern Recognition

2 Random Vectors and their Properties 41

I
Fortunately, in pattern recognition problems, I X is rarely computed
directly. Instead, In IC I is commonly used, which can be computed from the
eigenvalues as
n
InICI = Zlnh,. (2.152)
i=l

Fortheaboveexample,InlCI =C,!!, Inh, +90ln (O.l/9O)=C,!!, Inhi -612.2.
As far as the inverse is concerned, each element of I:-' is given by the ratio
of a cofactor (the determinant of an (n -l)x(n - 1) matrix B ) and I X I . The cofactor
is the product of (n -1) eigenvalues of B, while I I is the product of n eigenvalues
of Z. Assuming that (n -I) eigenvalues of the denominator are, roughly speaking,
cancelled out with (n -1) eigenvalues of the numerator, I B I / I C I is proportional to
l/hk where hk is one of the eigenvalues of X. Therefore, although I C I becomes
extremely small as the above example indicates, each element of C-l does not go
up to an extremely large number. In order to avoid I B I / I I: I = 010 in computation,
it is suggested to use the following formula to compute the inverse matrix.

(2.153)

Again, the eigenvalues and eigenvectors of C are computed first, and then C-' is
obtained by (2.153). Recall from (2.129) that, if A and @ are the eigenvalue and
eigenvector matrices of C, A-I and 0 are the eigenvalue and eigenvector matrices
of C-' . Also, any matrix Q can be expressed by (2.138), using the eigenvalues and
eigenvectors.

Matrix Inversion

Diagonalization of matrices is particularly useful when we need the inverse
of matrices.
From (2.66), a distance function is expressed by
d$(X) = (X - M)'Z-'(X - M) = (Y - D)'A-l(Y - D)

(2.154)

where D = [d, . . . d,,]' and A are the expected vector and diagonal covariance