Page 62 - Introduction to Statistical Pattern Recognition

44 Introduction to Statistical Pattern Recognition

0
1
0

1 (2.165)
0
0

0

I' n-r-
Therefore, Q" is the inverse matrix of Q in the subspace spanned by I' eigenvec-
tors, and satisfies
QQ*Q =e. (2.166)

Generalized inverse: Equation (2.166) suggests a general way to define the
"inverse" of a rectangular (not square) singular matrix [IO]. The generalized
inverse of an m x n matrix R of rank r is an n x m matrix R# satisfying

RR~R =R , (2.167)
o# =or. (2.168)

The column vectors of R are seen to be eigenvectors of the m x m matrix (RR' ),
among which the r's are linearly independent with eigenvalues equal to 1. Also,
(rn -r) eigenvalues of (RR') must be zero. The matrix (RR') has the properties
of a projection matrix and is useful in linear regression analysis [6].
A particular form of R# is most often used. Let B be an m x r matrix whose
columns are the linearly independent columns of R. Then R can be expressed by