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One of the decompositions mentioned in the Wikipedia page on matrix decompositions is the algebraic polar decomposition. This factors a square complex matrix $M$ into $M = SQ$ where $S = S^T$ and $QQ^T = I$. This is similar to the usual polar decomposition, except that in the usual case $S = \overline S^T$ and $Q\overline Q^T = I$.

In a 1990 paper by Kaplansky, it is shown that this decomposition exists if and only if $M^T M$ is similar to $M M^T$, and in particular this is true for all invertible matrices.

Does this decomposition have any known applications? I guess an obstacle to applications is that the complex orthogonal matrices don't form a compact space.

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The "algebraic polar decomposition" was introduced pre-Kaplansky by Choudhury and Horn as a A Complex Orthogonal-Symmetric Analog of the Polar Decomposition (1987).

An application to improve the convergence of iterative algorithms has been developed in A coupled joint eigenvalue decomposition algorithm for canonical polyadic decomposition of tensors (2016).

In this paper we propose a novel algorithm to compute the joint eigenvalue decomposition of a set of squares matrices. This problem is at the heart of recent direct canonical polyadic decomposition algorithms. Contrary to the existing approaches the proposed algorithm can deal equally with real or complex-valued matrices without any modifications. The algorithm is based on the algebraic polar decomposition which allows to make the optimization step directly with complex parameters. Furthermore, both factorization matrices are estimated jointly. This “coupled” approach allows us to limit the numerical complexity of the algorithm.

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  • $\begingroup$ Thank you for this! $\endgroup$
    – wlad
    Feb 12 at 23:15

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