[Merged by Bors] - feat(LinearAlgebra/Matrix): Woodbury Identity #16325

4hma4d · 2024-08-30T19:17:14Z

This adds the Woodbury Identity.

Zulip discussion

Also corrects some bad deprecations introduced in #16590, which affected development of this PR.

Co-authored-by: Mohanad Ahmad

github-actions · 2024-08-30T19:17:54Z

PR summary d51e3a90ee

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Data.Matrix.Invertible	803	805	+2 (+0.25%)

Import changes for all files

Files	Import difference
`Mathlib.Data.Matrix.Invertible`	2

Declarations diff

+ add_mul_mul_invOf_mul_eq_one
+ add_mul_mul_invOf_mul_eq_one'
+ add_mul_mul_inv_eq_sub
+ invOf_add_mul_mul
+ invertibleAddMulMul

You can run this locally as follows

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

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean

vihdzp · 2024-09-01T20:21:39Z

Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean

+variable (A: Matrix n n α) (U : Matrix n m α) (C : Matrix m m α) (V : Matrix m n α)
+variable [Invertible A] [Invertible C][Invertible (C⁻¹ + V * A⁻¹* U)]
+
+lemma add_mul_mul_inv_mul_eq_one: (A+U*C*V)*(A⁻¹-A⁻¹*U*(C⁻¹+V*A⁻¹*U)⁻¹*V*A⁻¹) = 1 := by


Suggested change

lemma add_mul_mul_inv_mul_eq_one: (A+U*C*V)*(A⁻¹-A⁻¹*U*(C⁻¹+V*A⁻¹*U)⁻¹*V*A⁻¹) = 1 := by

lemma add_mul_mul_inv_mul_eq_one: (A + U * C * V) * (A⁻¹ - A⁻¹ * U * (C⁻¹ + V * A⁻¹ * U)⁻¹ * V * A⁻¹) = 1 := by

Can you also properly space out the rest of the proof?

I think it might be worth deviating from the "spaces aroung *" style here, and only adding them around the additive operators.

Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean

Co-authored-by: Violeta Hernández <[email protected]>

Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean

…nto woodbury

eric-wieser · 2024-09-09T09:55:35Z

Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean

+variable (A: Matrix n n α) (U : Matrix n m α) (C : Matrix m m α) (V : Matrix m n α)
+variable [Invertible A] [Invertible C] [Invertible (⅟C + V * ⅟A* U)]
+
+lemma add_mul_mul_invOf_mul_eq_one : (A + U*C*V)*(⅟A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A) = 1 := by


Can you move this to Matrix/Invertible.lean, and generalize it from CommRing to Ring?

Should i move all of it or just this lemma?

Yes, all the lemmas currently in this PR. The lemma I suggest below about inv would remain in this file.

@4hma4d, do you want me to do the move for you? Happy to do so to get this PR over the line, but don't want to tread on your toes.

Sorry for the delay. I didn't move add_mul_mul_inv since it relies on Matrix.invertibleOfRightInverse which is proved in NonsingularInverse

I didn't move add_mul_mul_inv since it relies on Matrix.invertibleOfRightInverse which is proved in NonsingularInverse

Indeed, to prove add_mul_mul_inv for the non-commutative case you have to repeat the whole proof, but with multiplication on the left instead! I've pushed a commit that does that.

Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean

Mathlib/Data/Matrix/Invertible.lean

eric-wieser · 2024-09-22T22:03:00Z

Mathlib/Data/Matrix/Invertible.lean

+-- as above, but with multiplication reversed
+lemma add_mul_mul_invOf_mul_eq_one' :
+    (⅟A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A)*(A + U*C*V) = 1 := by
+  calc
+    (⅟A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A)*(A + U*C*V)
+    _ = ⅟A*A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A*A + ⅟A*U*C*V - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A*U*C*V := by
+      simp_rw [add_sub_assoc, _root_.mul_add, _root_.sub_mul, Matrix.mul_assoc]
+    _ = (1 + ⅟A*U*C*V) - (⅟A*U*⅟(⅟C + V*⅟A*U)*V + ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A*U*C*V) := by
+      rw [invOf_mul_self, Matrix.mul_invOf_mul_self_cancel]
+      abel
+    _ = 1 + ⅟A*U*C*V - ⅟A*U*⅟(⅟C + V*⅟A*U)*(V + V*⅟A*U*C*V) := by
+      rw [sub_right_inj, Matrix.mul_add]
+      simp_rw [Matrix.mul_assoc]
+    _ = 1 + ⅟A*U*C*V - ⅟A*U*⅟(⅟C + V*⅟A*U)*(⅟C + V*⅟A*U)*C*V := by
+      congr 1
+      simp only [Matrix.mul_add, Matrix.add_mul, ← Matrix.mul_assoc,
+        Matrix.mul_invOf_mul_self_cancel]
+    _ = 1 := by
+      rw [Matrix.mul_invOf_mul_self_cancel]
+      abel


I copy-pasted this from your proof above, and tweaked it to reverse all the multiplications. Thanks for writing the original which made this relatively easy!

4hma4d · 2024-09-24T20:07:59Z

Is this fine? I added the spaces in Nonsingularinverse since I don't think it affects clarity there, and I added a comment in Invertible

eric-wieser · 2024-09-25T13:44:13Z

maintainer merge

Let's let another maintainer take the final look.
Thanks for your work here @4hma4d!

github-actions · 2024-09-25T13:44:39Z

🚀 Pull request has been placed on the maintainer queue by eric-wieser.

riccardobrasca · 2024-09-25T13:52:38Z

Thanks!

bors merge

This adds the [Woodbury Identity](https://en.wikipedia.org/wiki/Woodbury_matrix_identity). [Zulip discussion](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Woodbury.20identity/near/462245284) Co-authored-by: Mohanad Ahmad Co-authored-by: Eric Wieser <[email protected]>

mathlib-bors · 2024-09-25T13:54:43Z

Build failed:

Build

riccardobrasca · 2024-09-25T13:58:05Z

Can you please merge master and fix the problem? I think it is just something that is now deprecated.

bors d+

mathlib-bors · 2024-09-25T13:58:09Z

✌️ 4hma4d can now approve this pull request. To approve and merge a pull request, simply reply with bors r+. More detailed instructions are available here.

4hma4d · 2024-09-26T03:23:56Z

bors r+

This adds the [Woodbury Identity](https://en.wikipedia.org/wiki/Woodbury_matrix_identity). [Zulip discussion](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Woodbury.20identity/near/462245284) Also corrects some bad deprecations introduced in #16590, which affected development of this PR. Co-authored-by: Mohanad Ahmad Co-authored-by: Eric Wieser <[email protected]>

mathlib-bors · 2024-09-26T03:43:39Z

Pull request successfully merged into master.

Build succeeded:

This was added to mathlib in leanprover-community/mathlib4#16325

This main result is `Matrix.add_mul_mul_inv_eq_sub` which was added to mathlib in leanprover-community/mathlib4#16325. Some of these other corollaries could probably be upstreamed.

This adds the [Woodbury Identity](https://en.wikipedia.org/wiki/Woodbury_matrix_identity). [Zulip discussion](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Woodbury.20identity/near/462245284) Also corrects some bad deprecations introduced in #16590, which affected development of this PR. Co-authored-by: Mohanad Ahmad Co-authored-by: Eric Wieser <[email protected]>

4hma4d added 4 commits August 25, 2024 00:13

Added Woodbury Identity

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update

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Fixed Spacing

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Co-authored-by: Violeta Hernández <[email protected]>

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Merge branch 'woodbury' of github.com:leanprover-community/mathlib4 i…

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…nto woodbury

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generalize the results to non-commutative rings

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swap Invertible for IsUnit

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eric-wieser changed the title ~~feat (LinearAlgebra/Matrix): Woodbury Identity~~ feat(LinearAlgebra/Matrix): Woodbury Identity Sep 25, 2024

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Merge remote-tracking branch 'origin/master' into woodbury

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fix the deprecations

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The woodbury identity and related results

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This was added to mathlib in leanprover-community/mathlib4#16325

eric-wieser added a commit to eric-wieser/lean-matrix-cookbook that referenced this pull request Sep 29, 2024

The woodbury identity and related results

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This was added to mathlib in leanprover-community/mathlib4#16325

eric-wieser added a commit to eric-wieser/lean-matrix-cookbook that referenced this pull request Sep 29, 2024

The woodbury identity and related results

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This was added to mathlib in leanprover-community/mathlib4#16325

eric-wieser added a commit to eric-wieser/lean-matrix-cookbook that referenced this pull request Sep 29, 2024

The woodbury identity and related results

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This was added to mathlib in leanprover-community/mathlib4#16325

eric-wieser added a commit to eric-wieser/lean-matrix-cookbook that referenced this pull request Sep 29, 2024

The woodbury identity and related results

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This was added to mathlib in leanprover-community/mathlib4#16325

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[Merged by Bors] - feat(LinearAlgebra/Matrix): Woodbury Identity #16325

[Merged by Bors] - feat(LinearAlgebra/Matrix): Woodbury Identity #16325

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edited by eric-wieser

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Import changes for modified files

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	lemma add_mul_mul_inv_mul_eq_one: (A+UCV)(A⁻¹-A⁻¹U(C⁻¹+VA⁻¹U)⁻¹V*A⁻¹) = 1 := by
	lemma add_mul_mul_inv_mul_eq_one: (A + U * C * V) * (A⁻¹ - A⁻¹ * U * (C⁻¹ + V * A⁻¹ * U)⁻¹ * V * A⁻¹) = 1 := by

[Merged by Bors] - feat(LinearAlgebra/Matrix): Woodbury Identity #16325

[Merged by Bors] - feat(LinearAlgebra/Matrix): Woodbury Identity #16325

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PR summary d51e3a90ee

Import changes for modified files

Declarations diff

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