linear_algebra.matrix.hermitian ⟷ Mathlib.LinearAlgebra.Matrix.Hermitian

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

@@ -323,11 +323,11 @@ theorem IsHermitian.adjugate [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA
 
 end CommRing
 
-section IsROrC
+section RCLike
 
-open IsROrC
+open RCLike
 
-variable [IsROrC α] [IsROrC β]
+variable [RCLike α] [RCLike β]
 
 #print Matrix.IsHermitian.coe_re_apply_self /-
 /-- The diagonal elements of a complex hermitian matrix are real. -/
@@ -364,7 +364,7 @@ theorem isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] {A : Matrix n n
 #align matrix.is_hermitian_iff_is_symmetric Matrix.isHermitian_iff_isSymmetric
 -/
 
-end IsROrC
+end RCLike
 
 end Matrix
 

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

@@ -3,7 +3,7 @@ Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
 -/
-import Mathbin.Analysis.InnerProductSpace.PiL2
+import Analysis.InnerProductSpace.PiL2
 
 #align_import linear_algebra.matrix.hermitian from "leanprover-community/mathlib"@"f60c6087a7275b72d5db3c5a1d0e19e35a429c0a"
 

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

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
-
-! This file was ported from Lean 3 source module linear_algebra.matrix.hermitian
-! leanprover-community/mathlib commit f60c6087a7275b72d5db3c5a1d0e19e35a429c0a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.InnerProductSpace.PiL2
 
+#align_import linear_algebra.matrix.hermitian from "leanprover-community/mathlib"@"f60c6087a7275b72d5db3c5a1d0e19e35a429c0a"
+
 /-! # Hermitian matrices
 
 > THIS FILE IS SYNCHRONIZED WITH MATHLIB4.

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -66,7 +66,7 @@ protected theorem IsHermitian.isSelfAdjoint {A : Matrix n n α} (h : A.IsHermiti
 #print Matrix.IsHermitian.ext /-
 @[ext]
 theorem IsHermitian.ext {A : Matrix n n α} : (∀ i j, star (A j i) = A i j) → A.IsHermitian := by
-  intro h; ext (i j); exact h i j
+  intro h; ext i j; exact h i j
 #align matrix.is_hermitian.ext Matrix.IsHermitian.ext
 -/
 
@@ -359,7 +359,7 @@ theorem isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] {A : Matrix n n
   · rintro (h : Aᴴ = A) x y
     rw [h]
   · intro h
-    ext (i j)
+    ext i j
     simpa only [(Pi.single_star i 1).symm, ← star_mul_vec, mul_one, dot_product_single,
       single_vec_mul, star_one, one_mul] using
       h (@Pi.single _ _ _ (fun i => AddZeroClass.toHasZero α) i 1)

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -36,7 +36,6 @@ variable {α β : Type _} {m n : Type _} {A : Matrix n n α}
 
 open scoped Matrix
 
--- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner α _ _ x y
 
 section Star
@@ -83,11 +82,13 @@ theorem IsHermitian.ext_iff {A : Matrix n n α} : A.IsHermitian ↔ ∀ i j, sta
 #align matrix.is_hermitian.ext_iff Matrix.IsHermitian.ext_iff
 -/
 
+#print Matrix.IsHermitian.map /-
 @[simp]
 theorem IsHermitian.map {A : Matrix n n α} (h : A.IsHermitian) (f : α → β)
     (hf : Function.Semiconj f star star) : (A.map f).IsHermitian :=
   (conjTranspose_map f hf).symm.trans <| h.Eq.symm ▸ rfl
 #align matrix.is_hermitian.map Matrix.IsHermitian.map
+-/
 
 #print Matrix.IsHermitian.transpose /-
 theorem IsHermitian.transpose {A : Matrix n n α} (h : A.IsHermitian) : Aᵀ.IsHermitian := by
@@ -108,17 +109,21 @@ theorem IsHermitian.conjTranspose {A : Matrix n n α} (h : A.IsHermitian) : Aᴴ
 #align matrix.is_hermitian.conj_transpose Matrix.IsHermitian.conjTranspose
 -/
 
+#print Matrix.IsHermitian.submatrix /-
 @[simp]
 theorem IsHermitian.submatrix {A : Matrix n n α} (h : A.IsHermitian) (f : m → n) :
     (A.submatrix f f).IsHermitian :=
   (conjTranspose_submatrix _ _ _).trans (h.symm ▸ rfl)
 #align matrix.is_hermitian.submatrix Matrix.IsHermitian.submatrix
+-/
 
+#print Matrix.isHermitian_submatrix_equiv /-
 @[simp]
 theorem isHermitian_submatrix_equiv {A : Matrix n n α} (e : m ≃ n) :
     (A.submatrix e e).IsHermitian ↔ A.IsHermitian :=
   ⟨fun h => by simpa using h.submatrix e.symm, fun h => h.submatrix _⟩
 #align matrix.is_hermitian_submatrix_equiv Matrix.isHermitian_submatrix_equiv
+-/
 
 end Star
 
@@ -133,6 +138,7 @@ theorem isHermitian_conjTranspose_iff (A : Matrix n n α) : Aᴴ.IsHermitian ↔
 #align matrix.is_hermitian_conj_transpose_iff Matrix.isHermitian_conjTranspose_iff
 -/
 
+#print Matrix.IsHermitian.fromBlocks /-
 /-- A block matrix `A.from_blocks B C D` is hermitian,
     if `A` and `D` are hermitian and `Bᴴ = C`. -/
 theorem IsHermitian.fromBlocks {A : Matrix m m α} {B : Matrix m n α} {C : Matrix n m α}
@@ -144,7 +150,9 @@ theorem IsHermitian.fromBlocks {A : Matrix m m α} {B : Matrix m n α} {C : Matr
   rw [from_blocks_conj_transpose]
   congr <;> assumption
 #align matrix.is_hermitian.from_blocks Matrix.IsHermitian.fromBlocks
+-/
 
+#print Matrix.isHermitian_fromBlocks_iff /-
 /-- This is the `iff` version of `matrix.is_hermitian.from_blocks`. -/
 theorem isHermitian_fromBlocks_iff {A : Matrix m m α} {B : Matrix m n α} {C : Matrix n m α}
     {D : Matrix n n α} :
@@ -154,6 +162,7 @@ theorem isHermitian_fromBlocks_iff {A : Matrix m m α} {B : Matrix m n α} {C :
       congr_arg toBlocks₂₂ h⟩,
     fun ⟨hA, hBC, hCB, hD⟩ => IsHermitian.fromBlocks hA hBC hD⟩
 #align matrix.is_hermitian_from_blocks_iff Matrix.isHermitian_fromBlocks_iff
+-/
 
 end InvolutiveStar
 
@@ -161,6 +170,7 @@ section AddMonoid
 
 variable [AddMonoid α] [StarAddMonoid α] [AddMonoid β] [StarAddMonoid β]
 
+#print Matrix.isHermitian_diagonal_of_self_adjoint /-
 /-- A diagonal matrix is hermitian if the entries are self-adjoint -/
 theorem isHermitian_diagonal_of_self_adjoint [DecidableEq n] (v : n → α) (h : IsSelfAdjoint v) :
     (diagonal v).IsHermitian :=
@@ -169,7 +179,9 @@ theorem isHermitian_diagonal_of_self_adjoint [DecidableEq n] (v : n → α) (h :
         v).trans <|
     congr_arg _ h
 #align matrix.is_hermitian_diagonal_of_self_adjoint Matrix.isHermitian_diagonal_of_self_adjoint
+-/
 
+#print Matrix.isHermitian_diagonal /-
 /-- A diagonal matrix is hermitian if the entries have the trivial `star` operation
 (such as on the reals). -/
 @[simp]
@@ -177,17 +189,22 @@ theorem isHermitian_diagonal [TrivialStar α] [DecidableEq n] (v : n → α) :
     (diagonal v).IsHermitian :=
   isHermitian_diagonal_of_self_adjoint _ (IsSelfAdjoint.all _)
 #align matrix.is_hermitian_diagonal Matrix.isHermitian_diagonal
+-/
 
+#print Matrix.isHermitian_zero /-
 @[simp]
 theorem isHermitian_zero : (0 : Matrix n n α).IsHermitian :=
   isSelfAdjoint_zero _
 #align matrix.is_hermitian_zero Matrix.isHermitian_zero
+-/
 
+#print Matrix.IsHermitian.add /-
 @[simp]
 theorem IsHermitian.add {A B : Matrix n n α} (hA : A.IsHermitian) (hB : B.IsHermitian) :
     (A + B).IsHermitian :=
   IsSelfAdjoint.add hA hB
 #align matrix.is_hermitian.add Matrix.IsHermitian.add
+-/
 
 end AddMonoid
 
@@ -195,13 +212,17 @@ section AddCommMonoid
 
 variable [AddCommMonoid α] [StarAddMonoid α]
 
+#print Matrix.isHermitian_add_transpose_self /-
 theorem isHermitian_add_transpose_self (A : Matrix n n α) : (A + Aᴴ).IsHermitian :=
   isSelfAdjoint_add_star_self A
 #align matrix.is_hermitian_add_transpose_self Matrix.isHermitian_add_transpose_self
+-/
 
+#print Matrix.isHermitian_transpose_add_self /-
 theorem isHermitian_transpose_add_self (A : Matrix n n α) : (Aᴴ + A).IsHermitian :=
   isSelfAdjoint_star_add_self A
 #align matrix.is_hermitian_transpose_add_self Matrix.isHermitian_transpose_add_self
+-/
 
 end AddCommMonoid
 
@@ -209,16 +230,20 @@ section AddGroup
 
 variable [AddGroup α] [StarAddMonoid α]
 
+#print Matrix.IsHermitian.neg /-
 @[simp]
 theorem IsHermitian.neg {A : Matrix n n α} (h : A.IsHermitian) : (-A).IsHermitian :=
   IsSelfAdjoint.neg h
 #align matrix.is_hermitian.neg Matrix.IsHermitian.neg
+-/
 
+#print Matrix.IsHermitian.sub /-
 @[simp]
 theorem IsHermitian.sub {A B : Matrix n n α} (hA : A.IsHermitian) (hB : B.IsHermitian) :
     (A - B).IsHermitian :=
   IsSelfAdjoint.sub hA hB
 #align matrix.is_hermitian.sub Matrix.IsHermitian.sub
+-/
 
 end AddGroup
 
@@ -226,29 +251,37 @@ section NonUnitalSemiring
 
 variable [NonUnitalSemiring α] [StarRing α] [NonUnitalSemiring β] [StarRing β]
 
+#print Matrix.isHermitian_mul_conjTranspose_self /-
 /-- Note this is more general than `is_self_adjoint.mul_star_self` as `B` can be rectangular. -/
 theorem isHermitian_mul_conjTranspose_self [Fintype n] (A : Matrix m n α) : (A ⬝ Aᴴ).IsHermitian :=
   by rw [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose]
 #align matrix.is_hermitian_mul_conj_transpose_self Matrix.isHermitian_mul_conjTranspose_self
+-/
 
+#print Matrix.isHermitian_transpose_mul_self /-
 /-- Note this is more general than `is_self_adjoint.star_mul_self` as `B` can be rectangular. -/
 theorem isHermitian_transpose_mul_self [Fintype m] (A : Matrix m n α) : (Aᴴ ⬝ A).IsHermitian := by
   rw [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose]
 #align matrix.is_hermitian_transpose_mul_self Matrix.isHermitian_transpose_mul_self
+-/
 
+#print Matrix.isHermitian_conjTranspose_mul_mul /-
 /-- Note this is more general than `is_self_adjoint.conjugate'` as `B` can be rectangular. -/
 theorem isHermitian_conjTranspose_mul_mul [Fintype m] {A : Matrix m m α} (B : Matrix m n α)
     (hA : A.IsHermitian) : (Bᴴ ⬝ A ⬝ B).IsHermitian := by
   simp only [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose, hA.eq,
     Matrix.mul_assoc]
 #align matrix.is_hermitian_conj_transpose_mul_mul Matrix.isHermitian_conjTranspose_mul_mul
+-/
 
+#print Matrix.isHermitian_mul_mul_conjTranspose /-
 /-- Note this is more general than `is_self_adjoint.conjugate` as `B` can be rectangular. -/
 theorem isHermitian_mul_mul_conjTranspose [Fintype m] {A : Matrix m m α} (B : Matrix n m α)
     (hA : A.IsHermitian) : (B ⬝ A ⬝ Bᴴ).IsHermitian := by
   simp only [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose, hA.eq,
     Matrix.mul_assoc]
 #align matrix.is_hermitian_mul_mul_conj_transpose Matrix.isHermitian_mul_mul_conjTranspose
+-/
 
 end NonUnitalSemiring
 
@@ -256,12 +289,14 @@ section Semiring
 
 variable [Semiring α] [StarRing α] [Semiring β] [StarRing β]
 
+#print Matrix.isHermitian_one /-
 /-- Note this is more general for matrices than `is_self_adjoint_one` as it does not
 require `fintype n`, which is necessary for `monoid (matrix n n R)`. -/
 @[simp]
 theorem isHermitian_one [DecidableEq n] : (1 : Matrix n n α).IsHermitian :=
   conjTranspose_one
 #align matrix.is_hermitian_one Matrix.isHermitian_one
+-/
 
 end Semiring
 
@@ -269,15 +304,19 @@ section CommRing
 
 variable [CommRing α] [StarRing α]
 
+#print Matrix.IsHermitian.inv /-
 theorem IsHermitian.inv [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian) :
     A⁻¹.IsHermitian := by simp [is_hermitian, conj_transpose_nonsing_inv, hA.eq]
 #align matrix.is_hermitian.inv Matrix.IsHermitian.inv
+-/
 
+#print Matrix.isHermitian_inv /-
 @[simp]
 theorem isHermitian_inv [Fintype m] [DecidableEq m] (A : Matrix m m α) [Invertible A] :
     A⁻¹.IsHermitian ↔ A.IsHermitian :=
   ⟨fun h => by rw [← inv_inv_of_invertible A]; exact is_hermitian.inv h, IsHermitian.inv⟩
 #align matrix.is_hermitian_inv Matrix.isHermitian_inv
+-/
 
 #print Matrix.IsHermitian.adjugate /-
 theorem IsHermitian.adjugate [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian) :
@@ -293,17 +332,22 @@ open IsROrC
 
 variable [IsROrC α] [IsROrC β]
 
+#print Matrix.IsHermitian.coe_re_apply_self /-
 /-- The diagonal elements of a complex hermitian matrix are real. -/
 theorem IsHermitian.coe_re_apply_self {A : Matrix n n α} (h : A.IsHermitian) (i : n) :
     (re (A i i) : α) = A i i := by rw [← conj_eq_iff_re, ← star_def, ← conj_transpose_apply, h.eq]
 #align matrix.is_hermitian.coe_re_apply_self Matrix.IsHermitian.coe_re_apply_self
+-/
 
+#print Matrix.IsHermitian.coe_re_diag /-
 /-- The diagonal elements of a complex hermitian matrix are real. -/
 theorem IsHermitian.coe_re_diag {A : Matrix n n α} (h : A.IsHermitian) :
     (fun i => (re (A.diag i) : α)) = A.diag :=
   funext h.coe_re_apply_self
 #align matrix.is_hermitian.coe_re_diag Matrix.IsHermitian.coe_re_diag
+-/
 
+#print Matrix.isHermitian_iff_isSymmetric /-
 /-- A matrix is hermitian iff the corresponding linear map is self adjoint. -/
 theorem isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] {A : Matrix n n α} :
     IsHermitian A ↔ A.toEuclideanLin.IsSymmetric :=
@@ -321,6 +365,7 @@ theorem isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] {A : Matrix n n
       h (@Pi.single _ _ _ (fun i => AddZeroClass.toHasZero α) i 1)
         (@Pi.single _ _ _ (fun i => AddZeroClass.toHasZero α) j 1)
 #align matrix.is_hermitian_iff_is_symmetric Matrix.isHermitian_iff_isSymmetric
+-/
 
 end IsROrC
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -91,7 +91,7 @@ theorem IsHermitian.map {A : Matrix n n α} (h : A.IsHermitian) (f : α → β)
 
 #print Matrix.IsHermitian.transpose /-
 theorem IsHermitian.transpose {A : Matrix n n α} (h : A.IsHermitian) : Aᵀ.IsHermitian := by
-  rw [is_hermitian, conj_transpose, transpose_map]; congr ; exact h
+  rw [is_hermitian, conj_transpose, transpose_map]; congr; exact h
 #align matrix.is_hermitian.transpose Matrix.IsHermitian.transpose
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module linear_algebra.matrix.hermitian
-! leanprover-community/mathlib commit caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c
+! leanprover-community/mathlib commit f60c6087a7275b72d5db3c5a1d0e19e35a429c0a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -12,6 +12,9 @@ import Mathbin.Analysis.InnerProductSpace.PiL2
 
 /-! # Hermitian matrices
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file defines hermitian matrices and some basic results about them.
 
 See also `is_self_adjoint`, which generalizes this definition to other star rings.
@@ -40,33 +43,45 @@ section Star
 
 variable [Star α] [Star β]
 
+#print Matrix.IsHermitian /-
 /-- A matrix is hermitian if it is equal to its conjugate transpose. On the reals, this definition
 captures symmetric matrices. -/
 def IsHermitian (A : Matrix n n α) : Prop :=
   Aᴴ = A
 #align matrix.is_hermitian Matrix.IsHermitian
+-/
 
+#print Matrix.IsHermitian.eq /-
 theorem IsHermitian.eq {A : Matrix n n α} (h : A.IsHermitian) : Aᴴ = A :=
   h
 #align matrix.is_hermitian.eq Matrix.IsHermitian.eq
+-/
 
+#print Matrix.IsHermitian.isSelfAdjoint /-
 protected theorem IsHermitian.isSelfAdjoint {A : Matrix n n α} (h : A.IsHermitian) :
     IsSelfAdjoint A :=
   h
 #align matrix.is_hermitian.is_self_adjoint Matrix.IsHermitian.isSelfAdjoint
+-/
 
+#print Matrix.IsHermitian.ext /-
 @[ext]
 theorem IsHermitian.ext {A : Matrix n n α} : (∀ i j, star (A j i) = A i j) → A.IsHermitian := by
   intro h; ext (i j); exact h i j
 #align matrix.is_hermitian.ext Matrix.IsHermitian.ext
+-/
 
+#print Matrix.IsHermitian.apply /-
 theorem IsHermitian.apply {A : Matrix n n α} (h : A.IsHermitian) (i j : n) : star (A j i) = A i j :=
   congr_fun (congr_fun h _) _
 #align matrix.is_hermitian.apply Matrix.IsHermitian.apply
+-/
 
+#print Matrix.IsHermitian.ext_iff /-
 theorem IsHermitian.ext_iff {A : Matrix n n α} : A.IsHermitian ↔ ∀ i j, star (A j i) = A i j :=
   ⟨IsHermitian.apply, IsHermitian.ext⟩
 #align matrix.is_hermitian.ext_iff Matrix.IsHermitian.ext_iff
+-/
 
 @[simp]
 theorem IsHermitian.map {A : Matrix n n α} (h : A.IsHermitian) (f : α → β)
@@ -74,18 +89,24 @@ theorem IsHermitian.map {A : Matrix n n α} (h : A.IsHermitian) (f : α → β)
   (conjTranspose_map f hf).symm.trans <| h.Eq.symm ▸ rfl
 #align matrix.is_hermitian.map Matrix.IsHermitian.map
 
+#print Matrix.IsHermitian.transpose /-
 theorem IsHermitian.transpose {A : Matrix n n α} (h : A.IsHermitian) : Aᵀ.IsHermitian := by
   rw [is_hermitian, conj_transpose, transpose_map]; congr ; exact h
 #align matrix.is_hermitian.transpose Matrix.IsHermitian.transpose
+-/
 
+#print Matrix.isHermitian_transpose_iff /-
 @[simp]
 theorem isHermitian_transpose_iff (A : Matrix n n α) : Aᵀ.IsHermitian ↔ A.IsHermitian :=
   ⟨by intro h; rw [← transpose_transpose A]; exact is_hermitian.transpose h, IsHermitian.transpose⟩
 #align matrix.is_hermitian_transpose_iff Matrix.isHermitian_transpose_iff
+-/
 
+#print Matrix.IsHermitian.conjTranspose /-
 theorem IsHermitian.conjTranspose {A : Matrix n n α} (h : A.IsHermitian) : Aᴴ.IsHermitian :=
   h.transpose.map _ fun _ => rfl
 #align matrix.is_hermitian.conj_transpose Matrix.IsHermitian.conjTranspose
+-/
 
 @[simp]
 theorem IsHermitian.submatrix {A : Matrix n n α} (h : A.IsHermitian) (f : m → n) :
@@ -105,10 +126,12 @@ section InvolutiveStar
 
 variable [InvolutiveStar α]
 
+#print Matrix.isHermitian_conjTranspose_iff /-
 @[simp]
 theorem isHermitian_conjTranspose_iff (A : Matrix n n α) : Aᴴ.IsHermitian ↔ A.IsHermitian :=
   IsSelfAdjoint.star_iff
 #align matrix.is_hermitian_conj_transpose_iff Matrix.isHermitian_conjTranspose_iff
+-/
 
 /-- A block matrix `A.from_blocks B C D` is hermitian,
     if `A` and `D` are hermitian and `Bᴴ = C`. -/
@@ -256,9 +279,11 @@ theorem isHermitian_inv [Fintype m] [DecidableEq m] (A : Matrix m m α) [Inverti
   ⟨fun h => by rw [← inv_inv_of_invertible A]; exact is_hermitian.inv h, IsHermitian.inv⟩
 #align matrix.is_hermitian_inv Matrix.isHermitian_inv
 
+#print Matrix.IsHermitian.adjugate /-
 theorem IsHermitian.adjugate [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian) :
     A.adjugate.IsHermitian := by simp [is_hermitian, adjugate_conj_transpose, hA.eq]
 #align matrix.is_hermitian.adjugate Matrix.IsHermitian.adjugate
+-/
 
 end CommRing
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -31,7 +31,7 @@ namespace Matrix
 
 variable {α β : Type _} {m n : Type _} {A : Matrix n n α}
 
-open Matrix
+open scoped Matrix
 
 -- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner α _ _ x y

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -56,11 +56,8 @@ protected theorem IsHermitian.isSelfAdjoint {A : Matrix n n α} (h : A.IsHermiti
 #align matrix.is_hermitian.is_self_adjoint Matrix.IsHermitian.isSelfAdjoint
 
 @[ext]
-theorem IsHermitian.ext {A : Matrix n n α} : (∀ i j, star (A j i) = A i j) → A.IsHermitian :=
-  by
-  intro h
-  ext (i j)
-  exact h i j
+theorem IsHermitian.ext {A : Matrix n n α} : (∀ i j, star (A j i) = A i j) → A.IsHermitian := by
+  intro h; ext (i j); exact h i j
 #align matrix.is_hermitian.ext Matrix.IsHermitian.ext
 
 theorem IsHermitian.apply {A : Matrix n n α} (h : A.IsHermitian) (i j : n) : star (A j i) = A i j :=
@@ -77,19 +74,13 @@ theorem IsHermitian.map {A : Matrix n n α} (h : A.IsHermitian) (f : α → β)
   (conjTranspose_map f hf).symm.trans <| h.Eq.symm ▸ rfl
 #align matrix.is_hermitian.map Matrix.IsHermitian.map
 
-theorem IsHermitian.transpose {A : Matrix n n α} (h : A.IsHermitian) : Aᵀ.IsHermitian :=
-  by
-  rw [is_hermitian, conj_transpose, transpose_map]
-  congr
-  exact h
+theorem IsHermitian.transpose {A : Matrix n n α} (h : A.IsHermitian) : Aᵀ.IsHermitian := by
+  rw [is_hermitian, conj_transpose, transpose_map]; congr ; exact h
 #align matrix.is_hermitian.transpose Matrix.IsHermitian.transpose
 
 @[simp]
 theorem isHermitian_transpose_iff (A : Matrix n n α) : Aᵀ.IsHermitian ↔ A.IsHermitian :=
-  ⟨by
-    intro h
-    rw [← transpose_transpose A]
-    exact is_hermitian.transpose h, IsHermitian.transpose⟩
+  ⟨by intro h; rw [← transpose_transpose A]; exact is_hermitian.transpose h, IsHermitian.transpose⟩
 #align matrix.is_hermitian_transpose_iff Matrix.isHermitian_transpose_iff
 
 theorem IsHermitian.conjTranspose {A : Matrix n n α} (h : A.IsHermitian) : Aᴴ.IsHermitian :=
@@ -262,9 +253,7 @@ theorem IsHermitian.inv [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.
 @[simp]
 theorem isHermitian_inv [Fintype m] [DecidableEq m] (A : Matrix m m α) [Invertible A] :
     A⁻¹.IsHermitian ↔ A.IsHermitian :=
-  ⟨fun h => by
-    rw [← inv_inv_of_invertible A]
-    exact is_hermitian.inv h, IsHermitian.inv⟩
+  ⟨fun h => by rw [← inv_inv_of_invertible A]; exact is_hermitian.inv h, IsHermitian.inv⟩
 #align matrix.is_hermitian_inv Matrix.isHermitian_inv
 
 theorem IsHermitian.adjugate [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module linear_algebra.matrix.hermitian
-! leanprover-community/mathlib commit 9abfa6f0727d5adc99067e325e15d1a9de17fd8e
+! leanprover-community/mathlib commit caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -281,7 +281,7 @@ variable [IsROrC α] [IsROrC β]
 
 /-- The diagonal elements of a complex hermitian matrix are real. -/
 theorem IsHermitian.coe_re_apply_self {A : Matrix n n α} (h : A.IsHermitian) (i : n) :
-    (re (A i i) : α) = A i i := by rw [← eq_conj_iff_re, ← star_def, ← conj_transpose_apply, h.eq]
+    (re (A i i) : α) = A i i := by rw [← conj_eq_iff_re, ← star_def, ← conj_transpose_apply, h.eq]
 #align matrix.is_hermitian.coe_re_apply_self Matrix.IsHermitian.coe_re_apply_self
 
 /-- The diagonal elements of a complex hermitian matrix are real. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/290a7ba01fbcab1b64757bdaa270d28f4dcede35

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module linear_algebra.matrix.hermitian
-! leanprover-community/mathlib commit 9f0d61b4475e3c3cba6636ab51cdb1f3949d2e1d
+! leanprover-community/mathlib commit 9abfa6f0727d5adc99067e325e15d1a9de17fd8e
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,8 @@ import Mathbin.Analysis.InnerProductSpace.PiL2
 
 This file defines hermitian matrices and some basic results about them.
 
+See also `is_self_adjoint`, which generalizes this definition to other star rings.
+
 ## Main definition
 
  * `matrix.is_hermitian` : a matrix `A : matrix n n α` is hermitian if `Aᴴ = A`.
@@ -34,9 +36,9 @@ open Matrix
 -- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner α _ _ x y
 
-section NonUnitalSemiring
+section Star
 
-variable [NonUnitalSemiring α] [StarRing α] [NonUnitalSemiring β] [StarRing β]
+variable [Star α] [Star β]
 
 /-- A matrix is hermitian if it is equal to its conjugate transpose. On the reals, this definition
 captures symmetric matrices. -/
@@ -48,6 +50,11 @@ theorem IsHermitian.eq {A : Matrix n n α} (h : A.IsHermitian) : Aᴴ = A :=
   h
 #align matrix.is_hermitian.eq Matrix.IsHermitian.eq
 
+protected theorem IsHermitian.isSelfAdjoint {A : Matrix n n α} (h : A.IsHermitian) :
+    IsSelfAdjoint A :=
+  h
+#align matrix.is_hermitian.is_self_adjoint Matrix.IsHermitian.isSelfAdjoint
+
 @[ext]
 theorem IsHermitian.ext {A : Matrix n n α} : (∀ i j, star (A j i) = A i j) → A.IsHermitian :=
   by
@@ -57,48 +64,13 @@ theorem IsHermitian.ext {A : Matrix n n α} : (∀ i j, star (A j i) = A i j) 
 #align matrix.is_hermitian.ext Matrix.IsHermitian.ext
 
 theorem IsHermitian.apply {A : Matrix n n α} (h : A.IsHermitian) (i j : n) : star (A j i) = A i j :=
-  by
-  unfold is_hermitian at h
-  rw [← h, conj_transpose_apply, star_star, h]
+  congr_fun (congr_fun h _) _
 #align matrix.is_hermitian.apply Matrix.IsHermitian.apply
 
 theorem IsHermitian.ext_iff {A : Matrix n n α} : A.IsHermitian ↔ ∀ i j, star (A j i) = A i j :=
   ⟨IsHermitian.apply, IsHermitian.ext⟩
 #align matrix.is_hermitian.ext_iff Matrix.IsHermitian.ext_iff
 
-theorem isHermitian_mul_conjTranspose_self [Fintype n] (A : Matrix n n α) : (A ⬝ Aᴴ).IsHermitian :=
-  by rw [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose]
-#align matrix.is_hermitian_mul_conj_transpose_self Matrix.isHermitian_mul_conjTranspose_self
-
-theorem isHermitian_transpose_mul_self [Fintype n] (A : Matrix n n α) : (Aᴴ ⬝ A).IsHermitian := by
-  rw [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose]
-#align matrix.is_hermitian_transpose_mul_self Matrix.isHermitian_transpose_mul_self
-
-theorem isHermitian_conjTranspose_mul_mul [Fintype m] {A : Matrix m m α} (B : Matrix m n α)
-    (hA : A.IsHermitian) : (Bᴴ ⬝ A ⬝ B).IsHermitian := by
-  simp only [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose, hA.eq,
-    Matrix.mul_assoc]
-#align matrix.is_hermitian_conj_transpose_mul_mul Matrix.isHermitian_conjTranspose_mul_mul
-
-theorem isHermitian_mul_mul_conjTranspose [Fintype m] {A : Matrix m m α} (B : Matrix n m α)
-    (hA : A.IsHermitian) : (B ⬝ A ⬝ Bᴴ).IsHermitian := by
-  simp only [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose, hA.eq,
-    Matrix.mul_assoc]
-#align matrix.is_hermitian_mul_mul_conj_transpose Matrix.isHermitian_mul_mul_conjTranspose
-
-theorem isHermitian_add_transpose_self (A : Matrix n n α) : (A + Aᴴ).IsHermitian := by
-  simp [is_hermitian, add_comm]
-#align matrix.is_hermitian_add_transpose_self Matrix.isHermitian_add_transpose_self
-
-theorem isHermitian_transpose_add_self (A : Matrix n n α) : (Aᴴ + A).IsHermitian := by
-  simp [is_hermitian, add_comm]
-#align matrix.is_hermitian_transpose_add_self Matrix.isHermitian_transpose_add_self
-
-@[simp]
-theorem isHermitian_zero : (0 : Matrix n n α).IsHermitian :=
-  conjTranspose_zero
-#align matrix.is_hermitian_zero Matrix.isHermitian_zero
-
 @[simp]
 theorem IsHermitian.map {A : Matrix n n α} (h : A.IsHermitian) (f : α → β)
     (hf : Function.Semiconj f star star) : (A.map f).IsHermitian :=
@@ -124,20 +96,6 @@ theorem IsHermitian.conjTranspose {A : Matrix n n α} (h : A.IsHermitian) : Aᴴ
   h.transpose.map _ fun _ => rfl
 #align matrix.is_hermitian.conj_transpose Matrix.IsHermitian.conjTranspose
 
-@[simp]
-theorem isHermitian_conjTranspose_iff (A : Matrix n n α) : Aᴴ.IsHermitian ↔ A.IsHermitian :=
-  ⟨by
-    intro h
-    rw [← conj_transpose_conj_transpose A]
-    exact is_hermitian.conj_transpose h, IsHermitian.conjTranspose⟩
-#align matrix.is_hermitian_conj_transpose_iff Matrix.isHermitian_conjTranspose_iff
-
-@[simp]
-theorem IsHermitian.add {A B : Matrix n n α} (hA : A.IsHermitian) (hB : B.IsHermitian) :
-    (A + B).IsHermitian :=
-  (conjTranspose_add _ _).trans (hA.symm ▸ hB.symm ▸ rfl)
-#align matrix.is_hermitian.add Matrix.IsHermitian.add
-
 @[simp]
 theorem IsHermitian.submatrix {A : Matrix n n α} (h : A.IsHermitian) (f : m → n) :
     (A.submatrix f f).IsHermitian :=
@@ -150,11 +108,16 @@ theorem isHermitian_submatrix_equiv {A : Matrix n n α} (e : m ≃ n) :
   ⟨fun h => by simpa using h.submatrix e.symm, fun h => h.submatrix _⟩
 #align matrix.is_hermitian_submatrix_equiv Matrix.isHermitian_submatrix_equiv
 
-/-- The real diagonal matrix `diagonal v` is hermitian. -/
+end Star
+
+section InvolutiveStar
+
+variable [InvolutiveStar α]
+
 @[simp]
-theorem isHermitian_diagonal [DecidableEq n] (v : n → ℝ) : (diagonal v).IsHermitian :=
-  diagonal_conjTranspose _
-#align matrix.is_hermitian_diagonal Matrix.isHermitian_diagonal
+theorem isHermitian_conjTranspose_iff (A : Matrix n n α) : Aᴴ.IsHermitian ↔ A.IsHermitian :=
+  IsSelfAdjoint.star_iff
+#align matrix.is_hermitian_conj_transpose_iff Matrix.isHermitian_conjTranspose_iff
 
 /-- A block matrix `A.from_blocks B C D` is hermitian,
     if `A` and `D` are hermitian and `Bᴴ = C`. -/
@@ -162,9 +125,7 @@ theorem IsHermitian.fromBlocks {A : Matrix m m α} {B : Matrix m n α} {C : Matr
     {D : Matrix n n α} (hA : A.IsHermitian) (hBC : Bᴴ = C) (hD : D.IsHermitian) :
     (A.fromBlocks B C D).IsHermitian :=
   by
-  have hCB : Cᴴ = B := by
-    rw [← hBC]
-    simp
+  have hCB : Cᴴ = B := by rw [← hBC, conj_transpose_conj_transpose]
   unfold Matrix.IsHermitian
   rw [from_blocks_conj_transpose]
   congr <;> assumption
@@ -180,35 +141,115 @@ theorem isHermitian_fromBlocks_iff {A : Matrix m m α} {B : Matrix m n α} {C :
     fun ⟨hA, hBC, hCB, hD⟩ => IsHermitian.fromBlocks hA hBC hD⟩
 #align matrix.is_hermitian_from_blocks_iff Matrix.isHermitian_fromBlocks_iff
 
-end NonUnitalSemiring
+end InvolutiveStar
 
-section Semiring
+section AddMonoid
 
-variable [Semiring α] [StarRing α] [Semiring β] [StarRing β]
+variable [AddMonoid α] [StarAddMonoid α] [AddMonoid β] [StarAddMonoid β]
 
+/-- A diagonal matrix is hermitian if the entries are self-adjoint -/
+theorem isHermitian_diagonal_of_self_adjoint [DecidableEq n] (v : n → α) (h : IsSelfAdjoint v) :
+    (diagonal v).IsHermitian :=
+  (-- TODO: add a `pi.has_trivial_star` instance and remove the `funext`
+        diagonal_conjTranspose
+        v).trans <|
+    congr_arg _ h
+#align matrix.is_hermitian_diagonal_of_self_adjoint Matrix.isHermitian_diagonal_of_self_adjoint
+
+/-- A diagonal matrix is hermitian if the entries have the trivial `star` operation
+(such as on the reals). -/
 @[simp]
-theorem isHermitian_one [DecidableEq n] : (1 : Matrix n n α).IsHermitian :=
-  conjTranspose_one
-#align matrix.is_hermitian_one Matrix.isHermitian_one
+theorem isHermitian_diagonal [TrivialStar α] [DecidableEq n] (v : n → α) :
+    (diagonal v).IsHermitian :=
+  isHermitian_diagonal_of_self_adjoint _ (IsSelfAdjoint.all _)
+#align matrix.is_hermitian_diagonal Matrix.isHermitian_diagonal
 
-end Semiring
+@[simp]
+theorem isHermitian_zero : (0 : Matrix n n α).IsHermitian :=
+  isSelfAdjoint_zero _
+#align matrix.is_hermitian_zero Matrix.isHermitian_zero
+
+@[simp]
+theorem IsHermitian.add {A B : Matrix n n α} (hA : A.IsHermitian) (hB : B.IsHermitian) :
+    (A + B).IsHermitian :=
+  IsSelfAdjoint.add hA hB
+#align matrix.is_hermitian.add Matrix.IsHermitian.add
 
-section Ring
+end AddMonoid
 
-variable [Ring α] [StarRing α] [Ring β] [StarRing β]
+section AddCommMonoid
+
+variable [AddCommMonoid α] [StarAddMonoid α]
+
+theorem isHermitian_add_transpose_self (A : Matrix n n α) : (A + Aᴴ).IsHermitian :=
+  isSelfAdjoint_add_star_self A
+#align matrix.is_hermitian_add_transpose_self Matrix.isHermitian_add_transpose_self
+
+theorem isHermitian_transpose_add_self (A : Matrix n n α) : (Aᴴ + A).IsHermitian :=
+  isSelfAdjoint_star_add_self A
+#align matrix.is_hermitian_transpose_add_self Matrix.isHermitian_transpose_add_self
+
+end AddCommMonoid
+
+section AddGroup
+
+variable [AddGroup α] [StarAddMonoid α]
 
 @[simp]
 theorem IsHermitian.neg {A : Matrix n n α} (h : A.IsHermitian) : (-A).IsHermitian :=
-  (conjTranspose_neg _).trans (congr_arg _ h)
+  IsSelfAdjoint.neg h
 #align matrix.is_hermitian.neg Matrix.IsHermitian.neg
 
 @[simp]
 theorem IsHermitian.sub {A B : Matrix n n α} (hA : A.IsHermitian) (hB : B.IsHermitian) :
     (A - B).IsHermitian :=
-  (conjTranspose_sub _ _).trans (hA.symm ▸ hB.symm ▸ rfl)
+  IsSelfAdjoint.sub hA hB
 #align matrix.is_hermitian.sub Matrix.IsHermitian.sub
 
-end Ring
+end AddGroup
+
+section NonUnitalSemiring
+
+variable [NonUnitalSemiring α] [StarRing α] [NonUnitalSemiring β] [StarRing β]
+
+/-- Note this is more general than `is_self_adjoint.mul_star_self` as `B` can be rectangular. -/
+theorem isHermitian_mul_conjTranspose_self [Fintype n] (A : Matrix m n α) : (A ⬝ Aᴴ).IsHermitian :=
+  by rw [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose]
+#align matrix.is_hermitian_mul_conj_transpose_self Matrix.isHermitian_mul_conjTranspose_self
+
+/-- Note this is more general than `is_self_adjoint.star_mul_self` as `B` can be rectangular. -/
+theorem isHermitian_transpose_mul_self [Fintype m] (A : Matrix m n α) : (Aᴴ ⬝ A).IsHermitian := by
+  rw [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose]
+#align matrix.is_hermitian_transpose_mul_self Matrix.isHermitian_transpose_mul_self
+
+/-- Note this is more general than `is_self_adjoint.conjugate'` as `B` can be rectangular. -/
+theorem isHermitian_conjTranspose_mul_mul [Fintype m] {A : Matrix m m α} (B : Matrix m n α)
+    (hA : A.IsHermitian) : (Bᴴ ⬝ A ⬝ B).IsHermitian := by
+  simp only [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose, hA.eq,
+    Matrix.mul_assoc]
+#align matrix.is_hermitian_conj_transpose_mul_mul Matrix.isHermitian_conjTranspose_mul_mul
+
+/-- Note this is more general than `is_self_adjoint.conjugate` as `B` can be rectangular. -/
+theorem isHermitian_mul_mul_conjTranspose [Fintype m] {A : Matrix m m α} (B : Matrix n m α)
+    (hA : A.IsHermitian) : (B ⬝ A ⬝ Bᴴ).IsHermitian := by
+  simp only [is_hermitian, conj_transpose_mul, conj_transpose_conj_transpose, hA.eq,
+    Matrix.mul_assoc]
+#align matrix.is_hermitian_mul_mul_conj_transpose Matrix.isHermitian_mul_mul_conjTranspose
+
+end NonUnitalSemiring
+
+section Semiring
+
+variable [Semiring α] [StarRing α] [Semiring β] [StarRing β]
+
+/-- Note this is more general for matrices than `is_self_adjoint_one` as it does not
+require `fintype n`, which is necessary for `monoid (matrix n n R)`. -/
+@[simp]
+theorem isHermitian_one [DecidableEq n] : (1 : Matrix n n α).IsHermitian :=
+  conjTranspose_one
+#align matrix.is_hermitian_one Matrix.isHermitian_one
+
+end Semiring
 
 section CommRing
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/02ba8949f486ebecf93fe7460f1ed0564b5e442c

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module linear_algebra.matrix.hermitian
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! leanprover-community/mathlib commit 9f0d61b4475e3c3cba6636ab51cdb1f3949d2e1d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -32,7 +32,7 @@ variable {α β : Type _} {m n : Type _} {A : Matrix n n α}
 open Matrix
 
 -- mathport name: «expr⟪ , ⟫»
-local notation "⟪" x ", " y "⟫" => @inner α (PiLp 2 fun _ : n => α) _ x y
+local notation "⟪" x ", " y "⟫" => @inner α _ _ x y
 
 section NonUnitalSemiring
 
@@ -251,14 +251,10 @@ theorem IsHermitian.coe_re_diag {A : Matrix n n α} (h : A.IsHermitian) :
 
 /-- A matrix is hermitian iff the corresponding linear map is self adjoint. -/
 theorem isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] {A : Matrix n n α} :
-    IsHermitian A ↔
-      LinearMap.IsSymmetric
-        ((PiLp.linearEquiv 2 α fun _ : n => α).symm.conj A.toLin' : Module.End α (PiLp 2 _)) :=
+    IsHermitian A ↔ A.toEuclideanLin.IsSymmetric :=
   by
   rw [LinearMap.IsSymmetric, (PiLp.equiv 2 fun _ : n => α).symm.Surjective.Forall₂]
-  simp only [LinearEquiv.conj_apply, LinearMap.comp_apply, LinearEquiv.coe_coe,
-    PiLp.linearEquiv_apply, PiLp.linearEquiv_symm_apply, LinearEquiv.symm_symm]
-  simp_rw [EuclideanSpace.inner_eq_star_dotProduct, Equiv.apply_symm_apply, to_lin'_apply,
+  simp only [to_euclidean_lin_pi_Lp_equiv_symm, EuclideanSpace.inner_piLp_equiv_symm, to_lin'_apply,
     star_mul_vec, dot_product_mul_vec]
   constructor
   · rintro (h : Aᴴ = A) x y

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

Quoting [@digama0](https://github.com/digama0):

These were actually never meant to go in the file, they are basically debugging information and only useful on significantly broken mathport files. You can safely remove all of them.

@@ -31,7 +31,6 @@ variable {α β : Type*} {m n : Type*} {A : Matrix n n α}
 
 open scoped Matrix
 
--- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner α _ _ x y
 
 section Star

These are stated in the most general form possible, where they simply unfold the definition. This allows a few proofs to be golfed.

@@ -43,6 +43,9 @@ captures symmetric matrices. -/
 def IsHermitian (A : Matrix n n α) : Prop := Aᴴ = A
 #align matrix.is_hermitian Matrix.IsHermitian
 
+instance (A : Matrix n n α) [Decidable (Aᴴ = A)] : Decidable (IsHermitian A) :=
+  inferInstanceAs <| Decidable (_ = _)
+
 theorem IsHermitian.eq {A : Matrix n n α} (h : A.IsHermitian) : Aᴴ = A := h
 #align matrix.is_hermitian.eq Matrix.IsHermitian.eq
 

IsROrC contains data, which goes against the expectation that classes prefixed with Is are prop-valued. People have been complaining about this on and off, so this PR renames IsROrC to RCLike.

@@ -269,11 +269,11 @@ theorem IsHermitian.zpow [Fintype m] [DecidableEq m] {A : Matrix m m α} (h : A.
 
 end CommRing
 
-section IsROrC
+section RCLike
 
-open IsROrC
+open RCLike
 
-variable [IsROrC α] [IsROrC β]
+variable [RCLike α] [RCLike β]
 
 /-- The diagonal elements of a complex hermitian matrix are real. -/
 theorem IsHermitian.coe_re_apply_self {A : Matrix n n α} (h : A.IsHermitian) (i : n) :
@@ -301,6 +301,6 @@ theorem isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] {A : Matrix n n
       single_vecMul, star_one, one_mul] using h (Pi.single i 1) (Pi.single j 1)
 #align matrix.is_hermitian_iff_is_symmetric Matrix.isHermitian_iff_isSymmetric
 
-end IsROrC
+end RCLike
 
 end Matrix

@@ -139,7 +139,7 @@ theorem isHermitian_diagonal_of_self_adjoint [DecidableEq n] (v : n → α) (h :
 #align matrix.is_hermitian_diagonal_of_self_adjoint Matrix.isHermitian_diagonal_of_self_adjoint
 
 /-- A diagonal matrix is hermitian if each diagonal entry is self-adjoint -/
-lemma isHermitian_diagonal_iff [DecidableEq n]{d : n → α} :
+lemma isHermitian_diagonal_iff [DecidableEq n] {d : n → α} :
     IsHermitian (diagonal d) ↔ (∀ i : n, IsSelfAdjoint (d i)) := by
   simp [isSelfAdjoint_iff, IsHermitian, conjTranspose, diagonal_transpose, diagonal_map]
 

This PR adds two main results about positive semidefinite matrices:

• a positive semidefinite matrix has a unique positive semidefinite square root;
• if A is positive semidefinite then x* A x = 0 implies A x = 0 (proof extracted from #8594).

Co-authored-by: Adrian Wüthrich (@awueth)

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -131,13 +131,18 @@ section AddMonoid
 
 variable [AddMonoid α] [StarAddMonoid α] [AddMonoid β] [StarAddMonoid β]
 
-/-- A diagonal matrix is hermitian if the entries are self-adjoint -/
+/-- A diagonal matrix is hermitian if the entries are self-adjoint (as a vector) -/
 theorem isHermitian_diagonal_of_self_adjoint [DecidableEq n] (v : n → α) (h : IsSelfAdjoint v) :
     (diagonal v).IsHermitian :=
   (-- TODO: add a `pi.has_trivial_star` instance and remove the `funext`
         diagonal_conjTranspose v).trans <| congr_arg _ h
 #align matrix.is_hermitian_diagonal_of_self_adjoint Matrix.isHermitian_diagonal_of_self_adjoint
 
+/-- A diagonal matrix is hermitian if each diagonal entry is self-adjoint -/
+lemma isHermitian_diagonal_iff [DecidableEq n]{d : n → α} :
+    IsHermitian (diagonal d) ↔ (∀ i : n, IsSelfAdjoint (d i)) := by
+  simp [isSelfAdjoint_iff, IsHermitian, conjTranspose, diagonal_transpose, diagonal_map]
+
 /-- A diagonal matrix is hermitian if the entries have the trivial `star` operation
 (such as on the reals). -/
 @[simp]

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
 -/
 import Mathlib.Analysis.InnerProductSpace.PiL2
+import Mathlib.LinearAlgebra.Matrix.ZPow
 
 #align_import linear_algebra.matrix.hermitian from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
 
@@ -215,6 +216,10 @@ theorem isHermitian_mul_mul_conjTranspose [Fintype m] {A : Matrix m m α} (B : M
   simp only [IsHermitian, conjTranspose_mul, conjTranspose_conjTranspose, hA.eq, Matrix.mul_assoc]
 #align matrix.is_hermitian_mul_mul_conj_transpose Matrix.isHermitian_mul_mul_conjTranspose
 
+lemma commute_iff [Fintype n] {A B : Matrix n n α}
+    (hA : A.IsHermitian) (hB : B.IsHermitian) : Commute A B ↔ (A * B).IsHermitian :=
+  hA.isSelfAdjoint.commute_iff hB.isSelfAdjoint
+
 end NonUnitalSemiring
 
 section Semiring
@@ -228,6 +233,9 @@ theorem isHermitian_one [DecidableEq n] : (1 : Matrix n n α).IsHermitian :=
   conjTranspose_one
 #align matrix.is_hermitian_one Matrix.isHermitian_one
 
+theorem IsHermitian.pow [Fintype n] [DecidableEq n] {A : Matrix n n α} (h : A.IsHermitian) (k : ℕ) :
+    (A ^ k).IsHermitian := IsSelfAdjoint.pow h _
+
 end Semiring
 
 section CommRing
@@ -248,6 +256,12 @@ theorem IsHermitian.adjugate [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA
     A.adjugate.IsHermitian := by simp [IsHermitian, adjugate_conjTranspose, hA.eq]
 #align matrix.is_hermitian.adjugate Matrix.IsHermitian.adjugate
 
+/-- Note that `IsSelfAdjoint.zpow` does not apply to matrices as they are not a division ring. -/
+theorem IsHermitian.zpow [Fintype m] [DecidableEq m] {A : Matrix m m α} (h : A.IsHermitian)
+    (k : ℤ) :
+    (A ^ k).IsHermitian := by
+  rw [IsHermitian, conjTranspose_zpow, h]
+
 end CommRing
 
 section IsROrC

This removes the (PiLp.equiv 2 fun i => α i) abbreviation, replacing it with its implementation (WithLp.equiv 2 (∀ i, α i)). The same thing is done for PiLp.linearEquiv.

@@ -270,7 +270,7 @@ theorem IsHermitian.coe_re_diag {A : Matrix n n α} (h : A.IsHermitian) :
 /-- A matrix is hermitian iff the corresponding linear map is self adjoint. -/
 theorem isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] {A : Matrix n n α} :
     IsHermitian A ↔ A.toEuclideanLin.IsSymmetric := by
-  rw [LinearMap.IsSymmetric, (PiLp.equiv 2 fun _ : n => α).symm.surjective.forall₂]
+  rw [LinearMap.IsSymmetric, (WithLp.equiv 2 (n → α)).symm.surjective.forall₂]
   simp only [toEuclideanLin_piLp_equiv_symm, EuclideanSpace.inner_piLp_equiv_symm, toLin'_apply,
     star_mulVec, dotProduct_mulVec]
   constructor

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).

@@ -194,24 +194,24 @@ section NonUnitalSemiring
 variable [NonUnitalSemiring α] [StarRing α] [NonUnitalSemiring β] [StarRing β]
 
 /-- Note this is more general than `IsSelfAdjoint.mul_star_self` as `B` can be rectangular. -/
-theorem isHermitian_mul_conjTranspose_self [Fintype n] (A : Matrix m n α) : (A ⬝ Aᴴ).IsHermitian :=
+theorem isHermitian_mul_conjTranspose_self [Fintype n] (A : Matrix m n α) : (A * Aᴴ).IsHermitian :=
   by rw [IsHermitian, conjTranspose_mul, conjTranspose_conjTranspose]
 #align matrix.is_hermitian_mul_conj_transpose_self Matrix.isHermitian_mul_conjTranspose_self
 
 /-- Note this is more general than `IsSelfAdjoint.star_mul_self` as `B` can be rectangular. -/
-theorem isHermitian_transpose_mul_self [Fintype m] (A : Matrix m n α) : (Aᴴ ⬝ A).IsHermitian := by
+theorem isHermitian_transpose_mul_self [Fintype m] (A : Matrix m n α) : (Aᴴ * A).IsHermitian := by
   rw [IsHermitian, conjTranspose_mul, conjTranspose_conjTranspose]
 #align matrix.is_hermitian_transpose_mul_self Matrix.isHermitian_transpose_mul_self
 
 /-- Note this is more general than `IsSelfAdjoint.conjugate'` as `B` can be rectangular. -/
 theorem isHermitian_conjTranspose_mul_mul [Fintype m] {A : Matrix m m α} (B : Matrix m n α)
-    (hA : A.IsHermitian) : (Bᴴ ⬝ A ⬝ B).IsHermitian := by
+    (hA : A.IsHermitian) : (Bᴴ * A * B).IsHermitian := by
   simp only [IsHermitian, conjTranspose_mul, conjTranspose_conjTranspose, hA.eq, Matrix.mul_assoc]
 #align matrix.is_hermitian_conj_transpose_mul_mul Matrix.isHermitian_conjTranspose_mul_mul
 
 /-- Note this is more general than `IsSelfAdjoint.conjugate` as `B` can be rectangular. -/
 theorem isHermitian_mul_mul_conjTranspose [Fintype m] {A : Matrix m m α} (B : Matrix n m α)
-    (hA : A.IsHermitian) : (B ⬝ A ⬝ Bᴴ).IsHermitian := by
+    (hA : A.IsHermitian) : (B * A * Bᴴ).IsHermitian := by
   simp only [IsHermitian, conjTranspose_mul, conjTranspose_conjTranspose, hA.eq, Matrix.mul_assoc]
 #align matrix.is_hermitian_mul_mul_conj_transpose Matrix.isHermitian_mul_mul_conjTranspose
 

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

This has nice performance benefits.

@@ -26,7 +26,7 @@ self-adjoint matrix, hermitian matrix
 
 namespace Matrix
 
-variable {α β : Type _} {m n : Type _} {A : Matrix n n α}
+variable {α β : Type*} {m n : Type*} {A : Matrix n n α}
 
 open scoped Matrix
 

• depends on: #5966

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

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
-
-! This file was ported from Lean 3 source module linear_algebra.matrix.hermitian
-! leanprover-community/mathlib commit caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.InnerProductSpace.PiL2
 
+#align_import linear_algebra.matrix.hermitian from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
+
 /-! # Hermitian matrices
 
 This file defines hermitian matrices and some basic results about them.

Co-authored-by: Xavier Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr> Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Pol'tta / Miyahara Kō <pol_tta@outlook.jp> Co-authored-by: Jason Yuen <jason_yuen2007@hotmail.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com> Co-authored-by: Jireh Loreaux <loreaujy@gmail.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com> Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com> Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

@@ -54,7 +54,7 @@ protected theorem IsHermitian.isSelfAdjoint {A : Matrix n n α} (h : A.IsHermiti
 
 -- @[ext] -- Porting note: incorrect ext, not a structure or a lemma proving x = y
 theorem IsHermitian.ext {A : Matrix n n α} : (∀ i j, star (A j i) = A i j) → A.IsHermitian := by
-  intro h; ext (i j); exact h i j
+  intro h; ext i j; exact h i j
 #align matrix.is_hermitian.ext Matrix.IsHermitian.ext
 
 theorem IsHermitian.apply {A : Matrix n n α} (h : A.IsHermitian) (i j : n) : star (A j i) = A i j :=
@@ -280,7 +280,7 @@ theorem isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] {A : Matrix n n
   · rintro (h : Aᴴ = A) x y
     rw [h]
   · intro h
-    ext (i j)
+    ext i j
     simpa only [(Pi.single_star i 1).symm, ← star_mulVec, mul_one, dotProduct_single,
       single_vecMul, star_one, one_mul] using h (Pi.single i 1) (Pi.single j 1)
 #align matrix.is_hermitian_iff_is_symmetric Matrix.isHermitian_iff_isSymmetric