Vol. 24, No. 1, 1968

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Products of positive definite matrices. II

Charles Ballantine

Vol. 24 (1968), No. 1, 7–17
Abstract

This paper is concerned with the problem of determining, for given positive integers n and j, which n×n matrices (of positive determinant) can be written as a product of j positive definite matrices. In §2 the 2 × 2 complex case is completely solved. In particular, it turns out that every 2 ×2 complex matrix of positive determinant can be factored into a product of five positive definite Hermitian matrices and, unless it is a negative scalar matrix, can even be written as a product of four positive definite matrices. Sections 3 and 4 deal with the general n × n case. In §3 it is shown that a scalar matrix λI can be written as a product of four positive definite Hermitian matrices only if the scalar λ is real and positive, and that λH ( λ complex, H Hermitian) can be written as a product of three positive definite matrices only if λH is itself positive definite. In §4 it is shown that every n × n real matrix of positive determinant can be written as a product of six positive definite real symmetric matrices and that every n × n complex matrix of positive determinant can be written as a product of eleven positive definite Hermitian matrices.

Mathematical Subject Classification
Primary: 15.60
Milestones
Received: 6 December 1966
Published: 1 January 1968
Authors
Charles Ballantine