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Pattern Recognition by Matthias Nagel, Matthias Richter, Jürgen Beyerer

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where E is a column-wise concatenation of the eigenvectors. Then the covariance matrix can be rewritten as

= i=1 d k i i i T =E E T .( 2.139 )

Because the eigenvectors constitute a orthonormal basis, E is orthogonal and ET = E−1. Therefore Equation (2.139) yields

= E T E.( 2.140 )

The Karhunen–Loève transformation of m is defined as

m ˜ _ = E T ( m _ μ ).( 2.141 )

The transformed random variable has zero mean

E{ m ˜ _ }=0( 2.142 )

and the covariance matrix is

Cov{ m ˜ _ }=E{ E T ( m _ μ ) T E }= E T E{ ( m

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