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3 12 Appendix C. Orthonormal Transformation

Conclusions:

- The linear and orthonormal transformation with matrix Z' does indeed
transform correlated features into uncorrelated ones.
- The squares of the eigenvalues of A are the variances in the new system of
coordinates.

It can also be shown that:

- The orthonormal transformation preserves the Mahalanobis distances, the matrix
A of the eigenvalues and the determinant of the covariance matrix.
Notice that the determinant of the covariance matrix has a physical
interpretation as the volume of the pattern cluster.
- The orthonormal transformation can also be performed with the transpose
matrix of the eigenvectors of C. This is precisely what is usually done, since in a
real problem we seldom know the matrix A. The only difference is that now the
eigenvalues themselves are the new variances.
For the example of section 2.3 (Figure 2.15). the new variances would be
A1=6.854 and &=0.1458.
We present two other interesting results:

Positive definiteness

Consider the quadratic form of a real and symmetric matrix C with all
eigenvalues positive:

Without loss of generality, we can assume the vectors y originated from an
orthonormal transformation of the vectors x: y = Zx. Thus:

This proves that d2 is positive for all non-null x.
Since covariance (and correlation) matrices are real symmetrical matrices with
positive eigenvalues (representing variances after the orthonormal transformation),
they are also positive definite.

Whitening transformation
Suppose that after applying the orthonormal transformation to the vectors y, as
expressed in (B-2). we apply another linear transformation using the matrix A-'.
The new covariance is: