A Complex Orthogonal-Symmetric Analog of the Polar Decomposition
Dipa Choudhury, Roger A. Horn · SIAM Journal on Algebraic and Discrete Methods · 1987
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.