Page 51 - Introduction to Statistical Pattern Recognition

2 Random Vectors and their Properties 33

or

be
can
By (2.951, (@-1’20T)-1 replaced by CI O@-1’2.

or
z~’z~(@@-’’~Y) (w-”~Y)A . (2.109)
=
Thus, the transformation matrix A = <D@-’/2yl is calculated as the eigenvector
matrix of^;'^^.
One fact should be mentioned here. The eigenvectors $i of a symmetric
matrix are orthogonal and satisfy @7qj =O for i #j. However, IT’& is not sym-
metric in general, and subsequently the eigenvectors ci are not mutually orthogo-
nal. Instead, the si’s are orthogonal with respect to CI : that is, <:XI cj = 0 for igj.
Furthermore, in order to make the 5,’s orthonormal with respect to XI to satisfy
the first equation of (2.101), the scale of si must be adjusted by such that
rT Y
(2.1 10)

Simultaneous diagonalization of two matrices is a very powerful tool in pat-
tern recognition, because many problems of pattern recognition consider two dis-
tributions for classification purposes. Also, there are many possible modifications
of the above discussion. These depend on what kind of properties we are
interested in, what kind of matrices are used, etc. In this section we will show one
of the modifications that will be used in later chapters.

Modification:

Theorem Let a matrix Q be given by a linear combination of two symmetric
matrices Q I and Q as

Q =aiQi +~2Q2 (2.1 11)
where a I and a2 are positive constants. If we normalize the eigenvectors with
respect to Q as the first equation of (2.101), Ql and Q2 will share the same