Principle-component analysis is concerned with explaining the va riance/covariance structure of any sample through the linear combinations such variables with the main goal in mind of dimension reduction. If data is multicollinear this is the main tactic to tackle that issue. A main theoretical underpinning is the following fact: Prove that a symmetric matrix has real eigenvalues and that the eigenvectors corresponding to distinct eigenvalues are mutually orthogonal. Use this fact to prove that any symmetric matrix A can be spectrally decomposed into CDC' with D a diagonal matrix containing the eigenvalues and C the normalized eigenvectors arranged column-wise