Principal Component Analysis

Say the reference cube is

,

where xi's are the flattened images, then the covariance matrix for this cube is

,

which is a Hermitian matrix with real eigenvalues (actually, it is real symmetric), since

.

Then the spectral decomposition of the covariance matrix becomes

,

where D = diag{λ1, λ2, ..., λn} is consisted of the eigenvalues, and Q is a real orthogonal matrix with the columns as the corresponding eigenvectors.

Assuming the diagonal entries follow λi ≥ λi+1, and rewrite the spectral decomposition as

,

then taking look at the components in the first square bracket, we get