Pattern Recognition

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

$\sum ={\displaystyle \sum _{i=1}^d{k}_i{\in }_i{\in }_i^T=E\wedge E^T.\left(2.139\right)}$

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

$\wedge =E^T\sum E.\left(2.140\right)$

The Karhunen–Loève transformation of m is defined as

$\underset{\_}{\tilde{m}}=E^T\left(\underset{\_}{m}-\text{μ}\right).\left(2.141\right)$

The transformed random variable has zero mean

$\text{E}\left\{\underset{\_}{\tilde{m}}\right\}=0\left(2.142\right)$

and the covariance matrix is

$\begin{array}{l}\text{Cov}\left\{\underset{\_}{\tilde{m}}\right\}=\text{E}\left\{E^T{\left(\underset{\_}{m}-\text{μ}\right)}^{T}E\right\}=E^T\text{E}\{(\underset{}{m}\end{array}$