Page 45 - Introduction to Statistical Pattern Recognition

2 Random Vectors and their Properties 27

is called an eigenvector-. When X is a symmetric n x n matrix, we haven real eigen-
values XI,. . ., h, and n real eigenvectors $,, . . . , @,. The eigenvectors
corresponding to two different eigenvalues are orthogonal. This can be proved as
follows: For h,, @, and h,, @,(A, #A,),

X@, =A,@, and XQJ = A,@, . (2.76)
Multiplying the first equation by @;, the second by $7, and subtracting the second
from the first gives

('1 - 'J)@T@, = @;'$I - = 9 (2.77)
since C is a symmetric matrix. Since h, +AJ,

(2.78)
OT@I = 0 '
Thus, (2.73), (2.74), and (2.78) can be rewritten as

(2.79)

(2.80)

where Q, is an n x n matrix, consisting of n eigenvectors as

@ = [@I ' ' ' 4411 (2.81)
and A is a diagonal matrix of eigenvalues as
F' 0

(2.82)

and I is the identity matrix. The matrices 0 and A will be called the eigenvector
matrix and the eigenvalue matrix, respectively.
Let us use @ as the transformation matrix A of (2.61) as
Y=OTX. (2.83)

Then, from (2.63),