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

$equation$

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

$equation$

The Karhunen–Loève transformation of m is defined as

$equation$

The transformed random variable has zero mean

$equation$

and the covariance matrix is

$equation$

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