If $U= \begin{pmatrix} I_{s} & C_{s\times r} \\ B_{r\times s} & -I_{r} \end{pmatrix},$ then $U= \begin{pmatrix} I & -C \\ 0 & I \end{pmatrix}\begin{pmatrix} I+CB & 0 \\ 0 & I \end{pmatrix}\begin{pmatrix} I & 0 \\ B & -I \end{pmatrix}.$ How do I get this factorization? Is there a method?
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This looks like a block matrix analog to https://en.wikipedia.org/wiki/LU_decomposition – Michael Gintz Oct 15 '20 at 19:46
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I'll try. If it works I'll type the answer. – Oct 15 '20 at 21:03