Principal component analysis with drop rank covariance matrix

Abstract

This paper considers the principal component analysis when the covariance matrix of the input vectors drops rank. This case sometimes happens when the total number of the input vectors is very limited. First, it is found that the eigen decomposition of the covariance matrix is not uniquely defined. This implies that different transform matrices could be obtained for performing the principal component analysis. Hence, the generalized form of the eigen decomposition of the covariance matrix is given. Also, it is found that the matrix with its columns being the eigenvectors of the covariance matrix is not necessary to be unitary. This implies that the transform for performing the principal component analysis may not be energy preserved. To address this issue, the necessary and sufficient condition for the matrix with its columns being the eigenvectors of the covariance matrix to be unitary is derived. Moreover, since the design of the unitary transform matrix for performing the principal component analysis is usually formulated as an optimization problem, the necessary and sufficient condition for the first order derivative of the Lagrange function to be equal to the zero vector is derived. In fact, the unitary matrix with its columns being the eigenvectors of the covariance matrix is only a particular case of the condition. Furthermore, the necessary and sufficient condition for the second order derivative of the Lagrange function to be a positive definite function is derived. It is found that the unitary matrix with its columns being the eigenvectors of the covariance matrix does not satisfy this condition. Computer numerical simulation results are given to valid the results.2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.