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.
