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3 8 2 Pattern Discrimination

After solving the homogeneous system of equations for the different eigenvalues
one obtains a family of eigenvectors z.

Figure 2.15. Eigenvectors of a linear transformation, maintaining the same
direction before and after the transformation. The standard deviations along the
eigenvectors are A, and A2.

Let us compute the eigenvalues for the transformation of Figure 2.11 :

For 2, the homogeneous system of equations is:

allowing us to choose the eigenvector: z, = [l 0.618]'. For A2 we can compute
the following eigenvector orthogonal to z,: z2 = [-1 1.6181' .
For a real symmetric and non-singular matrix A one always obtains d distinct
eigenvalues. Also, in this case, each pair of eigenvalues correspond to orthogonal
eigenvectors. Figure 2.15 illustrates the eigenvectors for the example we have been
solving. Using the eigenvectors one may obtain new features which are
uncorrelated. As a matter of fact, consider the matrix Z of unitary eigenvectors:

z=[z, z ?... Z',], (2-21)
with z,'z, = 0 and Z'Z = I (i.e., Z is an orthonormal matrix). (2-2 1 a)