# linear algebra – Are there any applications of the Algebraic Polar Decomposition?

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 $$Qoverline Q^T = I$$.

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.