Is it (always) possible to write a non-Hermitian, non-unitary square matrix as a product of A and B such that A is unitary (but may or may not be Hermitian) and B is Hermitian (but may or may not be non-unitary)? If not under what conditions it is possible?
2026-03-27 12:20:10.1774614010
Non-Hermitian, non-unitary square matrix factorized into unitary and Hermitian matrices
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Yes, this is always possible, analogous to writing every real (square) matrix as a product of orthogonal and symmetric matrices.
For invertible square matrices, these results are cases of Iwasawa decompositions, which hold quite generally for semi-simple and reductive real Lie groups (and also for semi-simple and $p$-adic groups!) The case of non-invertible matrices probably follows by taking a limit, but I'm not absolutely certain... The proofs for the concrete cases are not soooo hard, but not trivial. The abstracted, intrinsic case has much more baggage.