On a Multivariate Eigenvalue Problem, Part I: Algebraic Theory and a Power Method

Abstract

Multivariate eigenvalue problems for symmetric and positive definite matrices arise from multivariate statistics theory where coefficients are to be determined so that the resulting linear combinations of sets of random variables are maximally correlated. By using the method of Lagrange multipliers such an optimization problem can be reduced to the multivariate eigenvalue problem. For over 30 years an iterative method proposed by Horst [Psychometrika, 26 (1961), pp. 129–149J has been used for solving the multivariate eigenvalue problem. Yet the theory of convergence has never been complete. The number of solutions to the multivariate eigenvalue problem also remains unknown. This paper contains two new results. By using the degree theory, a closed form on the cardinality of solutions for the multivariate eigenvalue problem is first proved. A convergence property of Horst’s method by forming it as a generalization of the so-called power method is then proved. The discussion leads to new formulations of numerical methods.