Page 47 - Introduction to Statistical Pattern Recognition

2 Random Vectors and their Properties 29

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Fig. 2-1 Eigenvalues and eigenvectors of a distribution.

whitening process. The purpose of the second transformation is to change
the scales of the principal components in proportion to I/%. Figure 2-2 shows a
two-dimensional example.
A few properties of the whitening transformation are pointed out here as fol-
lows.
( 1) Whitening transformations are not orthonormal transformations because

(@A-l/z)T(@A-l/2) = A-I/2@T@A-I/Z = A-1 [ (2.90)
Therefore, Euclidean distances are not preserved:
llY112 = Y'Y = X'@A-'@'X = XTZilX # IIXII' . (2.91)

(2) After a whitening transformation, the covariance matrix is invariant
under any orthonormal transformation, because
YTIY = YTY =I . (2.92)

This property will be used for simultaneous diagonalization of two matrices.