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It's well known that LU decomposition is only numerically stable if it's combined with row and/or column pivoting. It makes me wonder if there are other matrix decompositions that can profitably be combined with pivoting, that are not simply special cases of LDU decomposition with block diagonal D.

More broadly, which matrix decompositions feature permutation matrices?

The motivation for this question is to gain a unified understanding of the features of matrix decompositions.

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    $\begingroup$ Question author here: It looks like the RRQR decomposition is an example $\endgroup$
    – wlad
    Commented Jan 18, 2021 at 23:01
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    $\begingroup$ en.wikipedia.org/wiki/Bruhat_decomposition $\endgroup$
    – Francois Ziegler
    Commented Jan 18, 2021 at 23:45
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    $\begingroup$ @FrancoisZiegler, isn't the LU decomposition just a translate of the Bruhat decomposition for $\operatorname{GL}_n$? $\endgroup$
    – LSpice
    Commented Jan 19, 2021 at 0:10
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    $\begingroup$ @LSpice I would say that LU decomposition is the largest piece of Bruhat decomposition. That it to say, Bruhat decomposition is $GL_n = \bigsqcup_{w \in S_n} L w U$. The $LU$ decomposition is the observation that the $w = \mathrm{Id}$ piece covers all but a piece of measure $0$. $\endgroup$
    – David E Speyer
    Commented Jan 19, 2021 at 2:13
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    $\begingroup$ I would say that LU and LUP decompositions are somewhat distinct from Bruhat decomposition for they work for singular matrices as well. $\endgroup$
    – Andrei Smolensky
    Commented Jan 19, 2021 at 11:40

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