Page 50 - Introduction to Statistical Pattern Recognition

32 Introduction to Statistical Pattern Recognition

(C 1
Fig. 2-3 Simultaneous diagonalization.

ATCIA =I and ATZ2A =A, (2.101)
where A and A are the eigenvalue and eigenvector matrices of X y ' C2.

ZT'C2A =AA . (2.102)

Proof Since A's are the eigenvalues of K from (2.98),
IK-hll =o. (2.103)

Replacing K and I by (2.95) and (2.96),
I o-'/*oT Z* - hC' I I I = 0 , (2.104)
I
I
Since the transformation matrix is nonsingular, I O-1/2@T # 0 and
I
I @O-112 # 0. Therefore,
I
Iz,-hCII =o or IC;'z2-hlI =o. (2.105)
Thus, h's are the eigenvalues of Cy1X2.
For the eigenvectors, inserting (2.96) into (2.98) yields