[Merged by Bors] - feat(linear_algebra/matrix/schur_complement): invertibility of block matrices #19156
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…ck matrices with an invertible corner
… matrices are invertible iff their diagonal is This follows the pattern used for `submatrix` and `diagonal` of setting up ```lean invertible (from_blocks A B 0 D) ≃ invertible A × invertible D ``` and using it to prove ```lean is_unit (from_blocks A B 0 D) ↔ is_unit A ∧ is_unit D ``` These results fall out as the special case of the general formula; though my hope is that they can also be used to prove it.
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| /-- A block matrix is invertible if the bottom right corner and the corresponding schur complement | ||
| is. -/ |
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Somewhere in this file you can add a cross-reference (and I guess in the other direction too!) to category_theory.preadditive.biproducts, which proves very similar results. I guess biprod.guassian and biprod.iso_elim are the most directly relevant ones.
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(That is in slightly more generality as it is about 2x2 matrices in a category rather than a ring, but whatever. That file also only does one direction of these invertibility results, and doesn't (I think) give the iff statements.)
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…matrices (#19156) Co-authored-by: Jeremy Tan Jie Rui <e0191785@u.nus.edu> Co-authored-by: Jireh Loreaux <loreaujy@gmail.com>
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Creating this for the purpose of having an easy diff against #18849