Is there a decomposition of $SL_n(\mathbb C)$ as a product of $O_n(\mathbb C)\times Sym_n(\mathbb C)$ ? I mean is there a result as the polar decomposition but with orthogonal (not unitary)?
thanks
2026-04-24 16:27:42.1777048062
Orthogonal complex matrices: polar decomposition
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Yes, this is known as the algebraic polar decomposition of a matrix. It is apparently equivalent to the similarity of $A^TA$ and $AA^T$ (when $A$ is an arbitrary complex matrix).
For a reference, see Choudhury & Horn 1987, pp. 219-225.