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There exists a polar decomposition for matrices over the reals. What I would like to know is if an analog has been shown for groups of matrices over finite fields. If not, it would be great to get some feedback as to how to proceed, or some reasons why it may not exist.

Specifically, I'm interested in decomposing invertible matrices over $\mathbb{F}_q$ as a product $x = s.o$ where $s$ is symmetric and $o$ is orthogonal.

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since the singular value decomposition does not generalize to finite fields, doesn't the same apply to the polar decomposition?… – Carlo Beenakker May 3 '13 at 8:14
Perhaps you should explain the motivation you have for such a decomposition. – Dima Pasechnik Sep 21 '13 at 7:55

Orthogonal group consists of fixed-points of an involutary automorphism in $GL(n)$. There is a general theory of involutions and symmetric varieties. For an involution $\sigma$ of a semi-simple algebraic group $G$, the subvariety $\{x\sigma(x)\mid x\in G\}$ is affine generalizes the notion set of symmetric matrices, viz. $ = GL(n)/O(n)$.

See works of Deconcni-Procesi or T.A. Springer for stratifying $G/H$ $H$ being the fixed-point subgroup of $\sigma$.

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