[Merged by Bors] - feat: lemmas about Matrix is symmetric iff bilinear form is symmetric

[gw90]

Adds 4 lemmas, all of which are variants of the statement that a matrix is symmetric if and only if the corresponding bilinear form is symmetric. These lemmas would help me in a downstream project.

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[github-actions]

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[github-actions]

PR summary 93894ffe7d

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ BilinForm.ext_iff_basis
+ LinearMap.BilinForm.isSymm_toMatrix'_iff_isSymm
+ LinearMap.BilinForm.isSymm_toMatrix_iff_isSymm
+ Matrix.isSymm_toBilin'_iff_isSymm
+ Matrix.isSymm_toBilin_iff_isSymm
+ ext_iff_basis

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 93894ff).

• +6 new declarations
• −0 removed declarations

+LinearMap.BilinForm.ext_iff_basis
+LinearMap.BilinForm.isSymm_toMatrix'_iff_isSymm
+LinearMap.BilinForm.isSymm_toMatrix_iff_isSymm
+LinearMap.ext_iff_basis
+Matrix.isSymm_toBilin'_iff_isSymm
+Matrix.isSymm_toBilin_iff_isSymm

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 93894ffe7d
Reference commit a7f9fbe434

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[themathqueen]

It's nice to see you contributing again!

(please remember to comment -awaiting-author to remove the label when you're ready for another review!)

[gw90]

-awaiting-author

[themathqueen]

Final thing, can you move LinearMap.BilinForm.ext_basis and LinearMap.BilinForm.sum_repr_mul_repr_mul to the bilinear file (where you put the iff version)? I don't think they belong in the Hom file?

(Or if you want, this can be in another PR)

[gw90]

Final thing, can you move LinearMap.BilinForm.ext_basis and LinearMap.BilinForm.sum_repr_mul_repr_mul to the bilinear file (where you put the iff version)? I don't think they belong in the Hom file?

(Or if you want, this can be in another PR)

I'd rather leave this as another PR, mainly because Mathlib.LinearAlgebra.BilinearForm.Hom also lists other to-dos about re-arranging/eliminating redundant code, and I think that deserves its own PR where everything is thought out all together.

[themathqueen]

Thanks!

maintainer merge

[github-actions]

🚀 Pull request has been placed on the maintainer queue by themathqueen.

[riccardobrasca]

Thanks!

bors merge

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

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