Every square complex matrix A can be factorized as $A = UH$, where U is unitary and H is positive semi-definite Hermitian. If A is nonsingular, it is known that one may write $A = QS$, where Q is complex orthogonal and S is complex symmetric. We develop necessary and sufficient conditions for there to be a factorization of this type when A is singular, and we give sufficient conditions for there to be at least one such factorization in which the factors commute.
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Choudhury et al. (1987) studied this question.
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