Page 43 - Introduction to Statistical Pattern Recognition
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2 Random Vectors and their Properties 25
Y=ATX, (2.61)
where A is an n x n matrix. Then, the expected vector and covariance matrix of Y
are
My=E(Y) =ATE(X] =ATMX, (2.62)
Xy = E ((Y - My)(Y - My)' ]
=A'E((X - M,y)(X - Mx)T}A
= A T ~ , X ~ (2.63)
where the following rule of matrices (matrices need not be square) is used
AB)^ = B ~ . A ~ (2.64)
A similar rule, which holds for the inversion of matrices, is
This time, the existence of (AB)-' ,A-', andB-' is required.
Example 3: The distance function of (2.22) for Y can be calculated as
d?(Y) = (Y - My)%;'(Y - M Y)
= (x - M,)~AA-'C,'(A~)-'A~(X - M~)
= (X - MX)%iI (X - M,)
= &(X) . (2.66)
That is, the distance of (2.22) is invariant under any nonsingular ( I A I t 0) linear
transformation.
Example 4: If X is normal with Mx and &, Y is also normal with MY and
Cy. Since the quadratic form in the exponential function is invariant, the density
function of Y is
