Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
feat: port Data.Matrix.Invertible (#5366)
Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>
- Loading branch information
1 parent
358faa8
commit e172d9e
Showing
3 changed files
with
117 additions
and
31 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -0,0 +1,114 @@ | ||
/- | ||
Copyright (c) 2023 Eric Wieser. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Eric Wieser | ||
! This file was ported from Lean 3 source module data.matrix.invertible | ||
! leanprover-community/mathlib commit 722b3b152ddd5e0cf21c0a29787c76596cb6b422 | ||
! Please do not edit these lines, except to modify the commit id | ||
! if you have ported upstream changes. | ||
-/ | ||
import Mathlib.Algebra.Invertible | ||
import Mathlib.Data.Matrix.Basic | ||
|
||
/-! # Extra lemmas about invertible matrices | ||
Many of the `Invertible` lemmas are about `*`; this restates them to be about `⬝`. | ||
For lemmas about the matrix inverse in terms of the determinant and adjugate, see `Matrix.inv` | ||
in `LinearAlgebra/Matrix/NonsingularInverse.lean`. | ||
-/ | ||
|
||
|
||
open scoped Matrix | ||
|
||
variable {m n : Type _} {α : Type _} | ||
|
||
variable [Fintype n] [DecidableEq n] [Semiring α] | ||
|
||
namespace Matrix | ||
|
||
/-- A copy of `invOf_mul_self` using `⬝` not `*`. -/ | ||
protected theorem invOf_mul_self (A : Matrix n n α) [Invertible A] : ⅟ A ⬝ A = 1 := | ||
invOf_mul_self A | ||
#align matrix.inv_of_mul_self Matrix.invOf_mul_self | ||
|
||
/-- A copy of `mul_invOf_self` using `⬝` not `*`. -/ | ||
protected theorem mul_invOf_self (A : Matrix n n α) [Invertible A] : A ⬝ ⅟ A = 1 := | ||
mul_invOf_self A | ||
#align matrix.mul_inv_of_self Matrix.mul_invOf_self | ||
|
||
/-- A copy of `invOf_mul_self_assoc` using `⬝` not `*`. -/ | ||
protected theorem invOf_mul_self_assoc (A : Matrix n n α) (B : Matrix n m α) [Invertible A] : | ||
⅟ A ⬝ (A ⬝ B) = B := by rw [← Matrix.mul_assoc, Matrix.invOf_mul_self, Matrix.one_mul] | ||
#align matrix.inv_of_mul_self_assoc Matrix.invOf_mul_self_assoc | ||
|
||
/-- A copy of `mul_invOf_self_assoc` using `⬝` not `*`. -/ | ||
protected theorem mul_invOf_self_assoc (A : Matrix n n α) (B : Matrix n m α) [Invertible A] : | ||
A ⬝ (⅟ A ⬝ B) = B := by rw [← Matrix.mul_assoc, Matrix.mul_invOf_self, Matrix.one_mul] | ||
#align matrix.mul_inv_of_self_assoc Matrix.mul_invOf_self_assoc | ||
|
||
/-- A copy of `mul_invOf_mul_self_cancel` using `⬝` not `*`. -/ | ||
protected theorem mul_invOf_mul_self_cancel (A : Matrix m n α) (B : Matrix n n α) [Invertible B] : | ||
A ⬝ ⅟ B ⬝ B = A := by rw [Matrix.mul_assoc, Matrix.invOf_mul_self, Matrix.mul_one] | ||
#align matrix.mul_inv_of_mul_self_cancel Matrix.mul_invOf_mul_self_cancel | ||
|
||
/-- A copy of `mul_mul_invOf_self_cancel` using `⬝` not `*`. -/ | ||
protected theorem mul_mul_invOf_self_cancel (A : Matrix m n α) (B : Matrix n n α) [Invertible B] : | ||
A ⬝ B ⬝ ⅟ B = A := by rw [Matrix.mul_assoc, Matrix.mul_invOf_self, Matrix.mul_one] | ||
#align matrix.mul_mul_inv_of_self_cancel Matrix.mul_mul_invOf_self_cancel | ||
|
||
/-- A copy of `invertibleMul` using `⬝` not `*`. -/ | ||
@[reducible] | ||
protected def invertibleMul (A B : Matrix n n α) [Invertible A] [Invertible B] : | ||
Invertible (A ⬝ B) := | ||
{ invertibleMul _ _ with invOf := ⅟ B ⬝ ⅟ A } | ||
#align matrix.invertible_mul Matrix.invertibleMul | ||
|
||
/-- A copy of `Invertible.mul` using `⬝` not `*`.-/ | ||
@[reducible] | ||
def _root_.Invertible.matrixMul {A B : Matrix n n α} (_ : Invertible A) (_ : Invertible B) : | ||
Invertible (A ⬝ B) := | ||
invertibleMul _ _ | ||
#align invertible.matrix_mul Invertible.matrixMul | ||
|
||
protected theorem invOf_mul {A B : Matrix n n α} [Invertible A] [Invertible B] | ||
[Invertible (A ⬝ B)] : ⅟ (A ⬝ B) = ⅟ B ⬝ ⅟ A := | ||
invOf_mul _ _ | ||
#align matrix.inv_of_mul Matrix.invOf_mul | ||
|
||
/-- A copy of `invertibleOfInvertibleMul` using `⬝` not `*`. -/ | ||
@[reducible] | ||
protected def invertibleOfInvertibleMul (a b : Matrix n n α) [Invertible a] [Invertible (a ⬝ b)] : | ||
Invertible b := | ||
{ invertibleOfInvertibleMul a b with invOf := ⅟ (a ⬝ b) ⬝ a } | ||
#align matrix.invertible_of_invertible_mul Matrix.invertibleOfInvertibleMul | ||
|
||
/-- A copy of `invertibleOfMulInvertible` using `⬝` not `*`. -/ | ||
@[reducible] | ||
protected def invertibleOfMulInvertible (a b : Matrix n n α) [Invertible (a ⬝ b)] [Invertible b] : | ||
Invertible a := | ||
{ invertibleOfMulInvertible a b with invOf := b ⬝ ⅟ (a ⬝ b) } | ||
#align matrix.invertible_of_mul_invertible Matrix.invertibleOfMulInvertible | ||
|
||
end Matrix | ||
|
||
/-- A copy of `Invertible.mulLeft` using `⬝` not `*`. -/ | ||
@[reducible] | ||
def Invertible.matrixMulLeft {a : Matrix n n α} (_ : Invertible a) (b : Matrix n n α) : | ||
Invertible b ≃ Invertible (a ⬝ b) where | ||
toFun _ := Matrix.invertibleMul a b | ||
invFun _ := Matrix.invertibleOfInvertibleMul a _ | ||
left_inv _ := Subsingleton.elim _ _ | ||
right_inv _ := Subsingleton.elim _ _ | ||
#align invertible.matrix_mul_left Invertible.matrixMulLeft | ||
|
||
/-- A copy of `Invertible.mulRight` using `⬝` not `*`. -/ | ||
@[reducible] | ||
def Invertible.matrixMulRight (a : Matrix n n α) {b : Matrix n n α} (_ : Invertible b) : | ||
Invertible a ≃ Invertible (a ⬝ b) where | ||
toFun _ := Matrix.invertibleMul a b | ||
invFun _ := Matrix.invertibleOfMulInvertible _ b | ||
left_inv _ := Subsingleton.elim _ _ | ||
right_inv _ := Subsingleton.elim _ _ | ||
#align invertible.matrix_mul_right Invertible.matrixMulRight |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters