data.matrix.invertible ⟷ Mathlib.Data.Matrix.Invertible

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -3,7 +3,7 @@ Copyright (c) 2023 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
-import Algebra.Invertible
+import Algebra.Invertible.Defs
 import Data.Matrix.Basic
 
 #align_import data.matrix.invertible from "leanprover-community/mathlib"@"5d0c76894ada7940957143163d7b921345474cbc"

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,8 +3,8 @@ Copyright (c) 2023 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
-import Mathbin.Algebra.Invertible
-import Mathbin.Data.Matrix.Basic
+import Algebra.Invertible
+import Data.Matrix.Basic
 
 #align_import data.matrix.invertible from "leanprover-community/mathlib"@"5d0c76894ada7940957143163d7b921345474cbc"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -28,116 +28,130 @@ variable [Fintype n] [DecidableEq n] [Semiring α]
 
 namespace Matrix
 
-#print Matrix.invOf_mul_self /-
+/- warning: matrix.inv_of_mul_self clashes with inv_of_mul_self -> invOf_mul_self
+Case conversion may be inaccurate. Consider using '#align matrix.inv_of_mul_self invOf_mul_selfₓ'. -/
+#print invOf_mul_self /-
 /-- A copy of `inv_of_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
+#align matrix.inv_of_mul_self invOf_mul_self
 -/
 
-#print Matrix.mul_invOf_self /-
+/- warning: matrix.mul_inv_of_self clashes with mul_inv_of_self -> mul_invOf_self
+Case conversion may be inaccurate. Consider using '#align matrix.mul_inv_of_self mul_invOf_selfₓ'. -/
+#print mul_invOf_self /-
 /-- A copy of `mul_inv_of_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
+#align matrix.mul_inv_of_self mul_invOf_self
 -/
 
 #print Matrix.invOf_mul_self_assoc /-
 /-- A copy of `inv_of_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]
+    ⅟ A ⬝ (A ⬝ B) = B := by rw [← Matrix.mul_assoc, invOf_mul_self, Matrix.one_mul]
 #align matrix.inv_of_mul_self_assoc Matrix.invOf_mul_self_assoc
 -/
 
 #print Matrix.mul_invOf_self_assoc /-
 /-- A copy of `mul_inv_of_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]
+    A ⬝ (⅟ A ⬝ B) = B := by rw [← Matrix.mul_assoc, mul_invOf_self, Matrix.one_mul]
 #align matrix.mul_inv_of_self_assoc Matrix.mul_invOf_self_assoc
 -/
 
 #print Matrix.mul_invOf_mul_self_cancel /-
 /-- A copy of `mul_inv_of_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]
+    A ⬝ ⅟ B ⬝ B = A := by rw [Matrix.mul_assoc, invOf_mul_self, Matrix.mul_one]
 #align matrix.mul_inv_of_mul_self_cancel Matrix.mul_invOf_mul_self_cancel
 -/
 
 #print Matrix.mul_mul_invOf_self_cancel /-
 /-- A copy of `mul_mul_inv_of_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]
+    A ⬝ B ⬝ ⅟ B = A := by rw [Matrix.mul_assoc, mul_invOf_self, Matrix.mul_one]
 #align matrix.mul_mul_inv_of_self_cancel Matrix.mul_mul_invOf_self_cancel
 -/
 
-#print Matrix.invertibleMul /-
+/- warning: matrix.invertible_mul clashes with invertible_mul -> invertibleMul
+Case conversion may be inaccurate. Consider using '#align matrix.invertible_mul invertibleMulₓ'. -/
+#print invertibleMul /-
 /-- A copy of `invertible_mul` 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
+#align matrix.invertible_mul invertibleMul
 -/
 
-#print Invertible.matrixMul /-
 /-- A copy of `invertible.mul` using `⬝` not `*`.-/
 @[reducible]
 def Invertible.matrixMul {A B : Matrix n n α} (ha : Invertible A) (hb : Invertible B) :
     Invertible (A ⬝ B) :=
   invertibleMul _ _
 #align invertible.matrix_mul Invertible.matrixMul
--/
 
-#print Matrix.invOf_mul /-
+/- warning: matrix.inv_of_mul clashes with inv_of_mul -> invOf_mul
+Case conversion may be inaccurate. Consider using '#align matrix.inv_of_mul invOf_mulₓ'. -/
+#print invOf_mul /-
 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
+#align matrix.inv_of_mul invOf_mul
 -/
 
-#print Matrix.invertibleOfInvertibleMul /-
+/- warning: matrix.invertible_of_invertible_mul clashes with invertible_of_invertible_mul -> invertibleOfInvertibleMul
+Case conversion may be inaccurate. Consider using '#align matrix.invertible_of_invertible_mul invertibleOfInvertibleMulₓ'. -/
+#print invertibleOfInvertibleMul /-
 /-- A copy of `invertible_of_invertible_mul` 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
+#align matrix.invertible_of_invertible_mul invertibleOfInvertibleMul
 -/
 
-#print Matrix.invertibleOfMulInvertible /-
+/- warning: matrix.invertible_of_mul_invertible clashes with invertible_of_mul_invertible -> invertibleOfMulInvertible
+Case conversion may be inaccurate. Consider using '#align matrix.invertible_of_mul_invertible invertibleOfMulInvertibleₓ'. -/
+#print invertibleOfMulInvertible /-
 /-- A copy of `invertible_of_mul_invertible` 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
+#align matrix.invertible_of_mul_invertible invertibleOfMulInvertible
 -/
 
 end Matrix
 
-#print Invertible.matrixMulLeft /-
+/- warning: invertible.matrix_mul_left clashes with invertible.mul_left -> Invertible.mulLeft
+Case conversion may be inaccurate. Consider using '#align invertible.matrix_mul_left Invertible.mulLeftₓ'. -/
+#print Invertible.mulLeft /-
 /-- A copy of `invertible.mul_left` using `⬝` not `*`. -/
 @[reducible]
-def Invertible.matrixMulLeft {a : Matrix n n α} (ha : Invertible a) (b : Matrix n n α) :
+def Invertible.mulLeft {a : Matrix n n α} (ha : Invertible a) (b : Matrix n n α) :
     Invertible b ≃ Invertible (a ⬝ b)
     where
-  toFun hb := Matrix.invertibleMul a b
-  invFun hab := Matrix.invertibleOfInvertibleMul a _
+  toFun hb := invertibleMul a b
+  invFun hab := invertibleOfInvertibleMul a _
   left_inv hb := Subsingleton.elim _ _
   right_inv hab := Subsingleton.elim _ _
-#align invertible.matrix_mul_left Invertible.matrixMulLeft
+#align invertible.matrix_mul_left Invertible.mulLeft
 -/
 
-#print Invertible.matrixMulRight /-
+/- warning: invertible.matrix_mul_right clashes with invertible.mul_right -> Invertible.mulRight
+Case conversion may be inaccurate. Consider using '#align invertible.matrix_mul_right Invertible.mulRightₓ'. -/
+#print Invertible.mulRight /-
 /-- A copy of `invertible.mul_right` using `⬝` not `*`. -/
 @[reducible]
-def Invertible.matrixMulRight (a : Matrix n n α) {b : Matrix n n α} (ha : Invertible b) :
+def Invertible.mulRight (a : Matrix n n α) {b : Matrix n n α} (ha : Invertible b) :
     Invertible a ≃ Invertible (a ⬝ b)
     where
-  toFun hb := Matrix.invertibleMul a b
-  invFun hab := Matrix.invertibleOfMulInvertible _ b
+  toFun hb := invertibleMul a b
+  invFun hab := invertibleOfMulInvertible _ b
   left_inv hb := Subsingleton.elim _ _
   right_inv hab := Subsingleton.elim _ _
-#align invertible.matrix_mul_right Invertible.matrixMulRight
+#align invertible.matrix_mul_right Invertible.mulRight
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,15 +2,12 @@
 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 5d0c76894ada7940957143163d7b921345474cbc
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.Invertible
 import Mathbin.Data.Matrix.Basic
 
+#align_import data.matrix.invertible from "leanprover-community/mathlib"@"5d0c76894ada7940957143163d7b921345474cbc"
+
 /-! # Extra lemmas about invertible matrices
 
 > THIS FILE IS SYNCHRONIZED WITH MATHLIB4.

mathlib commit https://github.com/leanprover-community/mathlib/commit/93f880918cb51905fd51b76add8273cbc27718ab

@@ -4,7 +4,7 @@ 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
+! leanprover-community/mathlib commit 5d0c76894ada7940957143163d7b921345474cbc
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.Data.Matrix.Basic
 
 /-! # Extra lemmas about invertible matrices
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 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.has_inv`

mathlib commit https://github.com/leanprover-community/mathlib/commit/6285167a053ad0990fc88e56c48ccd9fae6550eb

@@ -70,41 +70,52 @@ protected theorem mul_mul_invOf_self_cancel (A : Matrix m n α) (B : Matrix n n
 #align matrix.mul_mul_inv_of_self_cancel Matrix.mul_mul_invOf_self_cancel
 -/
 
+#print Matrix.invertibleMul /-
 /-- A copy of `invertible_mul` 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
+-/
 
+#print Invertible.matrixMul /-
 /-- A copy of `invertible.mul` using `⬝` not `*`.-/
 @[reducible]
 def Invertible.matrixMul {A B : Matrix n n α} (ha : Invertible A) (hb : Invertible B) :
     Invertible (A ⬝ B) :=
   invertibleMul _ _
 #align invertible.matrix_mul Invertible.matrixMul
+-/
 
+#print Matrix.invOf_mul /-
 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
+-/
 
+#print Matrix.invertibleOfInvertibleMul /-
 /-- A copy of `invertible_of_invertible_mul` 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
+-/
 
+#print Matrix.invertibleOfMulInvertible /-
 /-- A copy of `invertible_of_mul_invertible` 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
 
+#print Invertible.matrixMulLeft /-
 /-- A copy of `invertible.mul_left` using `⬝` not `*`. -/
 @[reducible]
 def Invertible.matrixMulLeft {a : Matrix n n α} (ha : Invertible a) (b : Matrix n n α) :
@@ -115,7 +126,9 @@ def Invertible.matrixMulLeft {a : Matrix n n α} (ha : Invertible a) (b : Matrix
   left_inv hb := Subsingleton.elim _ _
   right_inv hab := Subsingleton.elim _ _
 #align invertible.matrix_mul_left Invertible.matrixMulLeft
+-/
 
+#print Invertible.matrixMulRight /-
 /-- A copy of `invertible.mul_right` using `⬝` not `*`. -/
 @[reducible]
 def Invertible.matrixMulRight (a : Matrix n n α) {b : Matrix n n α} (ha : Invertible b) :
@@ -126,4 +139,5 @@ def Invertible.matrixMulRight (a : Matrix n n α) {b : Matrix n n α} (ha : Inve
   left_inv hb := Subsingleton.elim _ _
   right_inv hab := Subsingleton.elim _ _
 #align invertible.matrix_mul_right Invertible.matrixMulRight
+-/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/6285167a053ad0990fc88e56c48ccd9fae6550eb

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -26,7 +26,6 @@ in `LinearAlgebra/Matrix/NonsingularInverse.lean`.
 open scoped Matrix
 
 variable {m n : Type*} {α : Type*}
-
 variable [Fintype n] [DecidableEq n]
 
 namespace Matrix

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -89,8 +89,8 @@ variable [CommSemiring α] (A : Matrix n n α)
 /-- The transpose of an invertible matrix is invertible. -/
 instance invertibleTranspose [Invertible A] : Invertible Aᵀ where
   invOf := (⅟A)ᵀ
-  invOf_mul_self := by rw [←transpose_mul, mul_invOf_self, transpose_one]
-  mul_invOf_self := by rw [←transpose_mul, invOf_mul_self, transpose_one]
+  invOf_mul_self := by rw [← transpose_mul, mul_invOf_self, transpose_one]
+  mul_invOf_self := by rw [← transpose_mul, invOf_mul_self, transpose_one]
 #align matrix.invertible_transpose Matrix.invertibleTranspose
 
 lemma transpose_invOf [Invertible A] [Invertible Aᵀ] : (⅟A)ᵀ = ⅟(Aᵀ) := by
@@ -100,8 +100,8 @@ lemma transpose_invOf [Invertible A] [Invertible Aᵀ] : (⅟A)ᵀ = ⅟(Aᵀ) :
 /-- `Aᵀ` is invertible when `A` is. -/
 def invertibleOfInvertibleTranspose [Invertible Aᵀ] : Invertible A where
   invOf := (⅟(Aᵀ))ᵀ
-  invOf_mul_self := by rw [←transpose_one, ← mul_invOf_self Aᵀ, transpose_mul, transpose_transpose]
-  mul_invOf_self := by rw [←transpose_one, ← invOf_mul_self Aᵀ, transpose_mul, transpose_transpose]
+  invOf_mul_self := by rw [← transpose_one, ← mul_invOf_self Aᵀ, transpose_mul, transpose_transpose]
+  mul_invOf_self := by rw [← transpose_one, ← invOf_mul_self Aᵀ, transpose_mul, transpose_transpose]
 #align matrix.invertible__of_invertible_transpose Matrix.invertibleOfInvertibleTranspose
 
 /-- Together `Matrix.invertibleTranspose` and `Matrix.invertibleOfInvertibleTranspose` form an

@@ -62,7 +62,7 @@ protected theorem mul_mul_invOf_self_cancel (A : Matrix m n α) (B : Matrix n n
 #align matrix.invertible_of_invertible_mul invertibleOfInvertibleMul
 #align matrix.invertible_of_mul_invertible invertibleOfMulInvertible
 
-section conj_transpose
+section ConjTranspose
 variable [StarRing α] (A : Matrix n n α)
 
 /-- The conjugate transpose of an invertible matrix is invertible. -/
@@ -78,7 +78,7 @@ def invertibleOfInvertibleConjTranspose [Invertible Aᴴ] : Invertible A := by
 
 @[simp] lemma isUnit_conjTranspose : IsUnit Aᴴ ↔ IsUnit A := isUnit_star
 
-end conj_transpose
+end ConjTranspose
 
 end Semiring
 

Mathlib.Algebra.Invertible is used by fundamental tactics, and this essentially splits it into the part used by NormNum, and everything else.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -3,7 +3,6 @@ Copyright (c) 2023 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
-import Mathlib.Algebra.Invertible
 import Mathlib.Data.Matrix.Basic
 
 #align_import data.matrix.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"

The conjTranspose lemmas now work for non-commutative rings.

@@ -14,6 +14,13 @@ A few of the `Invertible` lemmas generalize to multiplication of rectangular mat
 
 For lemmas about the matrix inverse in terms of the determinant and adjugate, see `Matrix.inv`
 in `LinearAlgebra/Matrix/NonsingularInverse.lean`.
+
+## Main results
+
+* `Matrix.invertibleConjTranspose`
+* `Matrix.invertibleTranspose`
+* `Matrix.isUnit_conjTranpose`
+* `Matrix.isUnit_tranpose`
 -/
 
 
@@ -21,10 +28,13 @@ open scoped Matrix
 
 variable {m n : Type*} {α : Type*}
 
-variable [Fintype n] [DecidableEq n] [Semiring α]
+variable [Fintype n] [DecidableEq n]
 
 namespace Matrix
 
+section Semiring
+variable [Semiring α]
+
 #align matrix.inv_of_mul_self invOf_mul_self
 #align matrix.mul_inv_of_self mul_invOf_self
 
@@ -53,6 +63,63 @@ protected theorem mul_mul_invOf_self_cancel (A : Matrix m n α) (B : Matrix n n
 #align matrix.invertible_of_invertible_mul invertibleOfInvertibleMul
 #align matrix.invertible_of_mul_invertible invertibleOfMulInvertible
 
+section conj_transpose
+variable [StarRing α] (A : Matrix n n α)
+
+/-- The conjugate transpose of an invertible matrix is invertible. -/
+instance invertibleConjTranspose [Invertible A] : Invertible Aᴴ := Invertible.star _
+
+lemma conjTranspose_invOf [Invertible A] [Invertible Aᴴ] : (⅟A)ᴴ = ⅟(Aᴴ) := star_invOf _
+
+/-- A matrix is invertible if the conjugate transpose is invertible. -/
+def invertibleOfInvertibleConjTranspose [Invertible Aᴴ] : Invertible A := by
+  rw [← conjTranspose_conjTranspose A, ← star_eq_conjTranspose]
+  infer_instance
+#align matrix.invertible_of_invertible_conj_transpose Matrix.invertibleOfInvertibleConjTranspose
+
+@[simp] lemma isUnit_conjTranspose : IsUnit Aᴴ ↔ IsUnit A := isUnit_star
+
+end conj_transpose
+
+end Semiring
+
+section CommSemiring
+
+variable [CommSemiring α] (A : Matrix n n α)
+
+/-- The transpose of an invertible matrix is invertible. -/
+instance invertibleTranspose [Invertible A] : Invertible Aᵀ where
+  invOf := (⅟A)ᵀ
+  invOf_mul_self := by rw [←transpose_mul, mul_invOf_self, transpose_one]
+  mul_invOf_self := by rw [←transpose_mul, invOf_mul_self, transpose_one]
+#align matrix.invertible_transpose Matrix.invertibleTranspose
+
+lemma transpose_invOf [Invertible A] [Invertible Aᵀ] : (⅟A)ᵀ = ⅟(Aᵀ) := by
+  letI := invertibleTranspose A
+  convert (rfl : _ = ⅟(Aᵀ))
+
+/-- `Aᵀ` is invertible when `A` is. -/
+def invertibleOfInvertibleTranspose [Invertible Aᵀ] : Invertible A where
+  invOf := (⅟(Aᵀ))ᵀ
+  invOf_mul_self := by rw [←transpose_one, ← mul_invOf_self Aᵀ, transpose_mul, transpose_transpose]
+  mul_invOf_self := by rw [←transpose_one, ← invOf_mul_self Aᵀ, transpose_mul, transpose_transpose]
+#align matrix.invertible__of_invertible_transpose Matrix.invertibleOfInvertibleTranspose
+
+/-- Together `Matrix.invertibleTranspose` and `Matrix.invertibleOfInvertibleTranspose` form an
+equivalence, although both sides of the equiv are subsingleton anyway. -/
+@[simps]
+def transposeInvertibleEquivInvertible : Invertible Aᵀ ≃ Invertible A where
+  toFun := @invertibleOfInvertibleTranspose _ _ _ _ _ _
+  invFun := @invertibleTranspose _ _ _ _ _ _
+  left_inv _ := Subsingleton.elim _ _
+  right_inv _ := Subsingleton.elim _ _
+
+@[simp] lemma isUnit_transpose : IsUnit Aᵀ ↔ IsUnit A := by
+  simp only [← nonempty_invertible_iff_isUnit,
+    (transposeInvertibleEquivInvertible A).nonempty_congr]
+
+end CommSemiring
+
 end Matrix
 
 #align invertible.matrix_mul_left Invertible.mulLeft

The main difficulty here is that * has a slightly difference precedence to ⬝. notably around smul and neg.

The other annoyance is that ↑U ⬝ A ⬝ ↑U⁻¹ : Matrix m m 𝔸 now has to be written U.val * A * (U⁻¹).val in order to typecheck.

A downside of this change to consider: if you have a goal of A * (B * C) = (A * B) * C, mul_assoc now gives the illusion of matching, when in fact Matrix.mul_assoc is needed. Previously the distinct symbol made it easy to avoid this mistake.

On the flipside, there is now no need to rewrite by Matrix.mul_eq_mul all the time (indeed, the lemma is now removed).

@@ -10,7 +10,7 @@ import Mathlib.Data.Matrix.Basic
 
 /-! # Extra lemmas about invertible matrices
 
-Many of the `Invertible` lemmas are about `*`; this restates them to be about `⬝`.
+A few of the `Invertible` lemmas generalize to multiplication of rectangular matrices.
 
 For lemmas about the matrix inverse in terms of the determinant and adjugate, see `Matrix.inv`
 in `LinearAlgebra/Matrix/NonsingularInverse.lean`.
@@ -25,87 +25,35 @@ 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
+#align matrix.inv_of_mul_self invOf_mul_self
+#align matrix.mul_inv_of_self mul_invOf_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 `*`. -/
+/-- A copy of `invOf_mul_self_assoc` for rectangular matrices. -/
 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]
+    ⅟ A * (A * B) = B := by rw [← Matrix.mul_assoc, 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 `*`. -/
+/-- A copy of `mul_invOf_self_assoc` for rectangular matrices. -/
 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]
+    A * (⅟ A * B) = B := by rw [← Matrix.mul_assoc, 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 `*`. -/
+/-- A copy of `mul_invOf_mul_self_cancel` for rectangular matrices. -/
 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]
+    A * ⅟ B * B = A := by rw [Matrix.mul_assoc, 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 `*`. -/
+/-- A copy of `mul_mul_invOf_self_cancel` for rectangular matrices. -/
 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]
+    A * B * ⅟ B = A := by rw [Matrix.mul_assoc, 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
+#align matrix.invertible_mul invertibleMul
+#align matrix.inv_of_mul invOf_mul
+#align matrix.invertible_of_invertible_mul invertibleOfInvertibleMul
+#align matrix.invertible_of_mul_invertible 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
+#align invertible.matrix_mul_left Invertible.mulLeft
+#align invertible.matrix_mul_right Invertible.mulRight

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -19,7 +19,7 @@ in `LinearAlgebra/Matrix/NonsingularInverse.lean`.
 
 open scoped Matrix
 
-variable {m n : Type _} {α : Type _}
+variable {m n : Type*} {α : Type*}
 
 variable [Fintype n] [DecidableEq n] [Semiring α]
 

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,15 +2,12 @@
 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
 
+#align_import data.matrix.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
+
 /-! # Extra lemmas about invertible matrices
 
 Many of the `Invertible` lemmas are about `*`; this restates them to be about `⬝`.

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>