Some relations between the comparison of covariance matrices and principal component analysis

Abstract

While principal component analysis is a widely used technique in applied multivariate analysis, little attention is normally given to the comparison of covariance matrices. Based on Roy's largest and smallest roots' criterion, we expose some known properties of the eigenvectors of the matrix Σ 1 −1 Σ 2 . The linear combinations defined by these eigenvectors are discussed as a generalisation of principal component analysis of two groups, which can be useful in the case Σ 1 ≠ Σ 2 . The technique is illustrated by an example. A similar approach to the comparison of covariance matrices, based on the notion of Mahalanobis distance, is sketched. Finally, three equivalent conditions are given for the condition that two covariance matrices have identical principal axes. This leads to the definition of four degrees of similarity of two covariance matrices.