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Let $M$ be an invertible matrix. The Bruhat decomposition for the general linear group (https://en.wikipedia.org/wiki/Bruhat_decomposition) tells us that $M$ can be written as $M=U_1 w U_2$ with a unique permutation $w$. My question is whether there is a quick way to obtain this decomposition in Sage or at least the permutation $w$ associated to the matrix $M$?