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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Chu, Moody T. Watterson, J. Loren
Copyright Year	1993
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. 129149J 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 Horsts 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.
Starting Page	1089
Ending Page	1106
Page Count	18
File Format	PDF
ISSN	10648275
DOI	10.1137/0914066
e-ISSN	10957197
Issue Number	5
Volume Number	14
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-13
Access Restriction	Subscribed
Subject Keyword	Systems of equations power method Other matrix algorithms homotopy method Factor analysis and principal components; correspondence analysis Eigenvalues, eigenvectors multivariate eigenvalue problem
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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