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.