The ϕS polar decomposition when S is skew-symmetric

Abstract

Let $F$ be a field with characteristic not equal to 2, and $S\in M_{2n}\left(F\right)$ be skew-symmetric and nonsingular. For $X\in M_{2n}\left(F\right)$, we show that X has a ${\varphi }_S$ polar decomposition if and only if (a) ${\varphi }_S\left(X\right)X$ has a ${\varphi }_S$-symmetric square root, (b) $X{\varphi }_S\left(X\right)$ is similar to ${\varphi }_S\left(X\right)X$, and (c) rank ${\left[X{\varphi }_S\right(X\left)\right]}^{t}X$ is even for all nonnegative integers t.