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

@@ -51,7 +51,7 @@ self-adjoint operator, spectral theorem, diagonalization theorem
 -/
 
 
-variable {𝕜 : Type _} [IsROrC 𝕜] [dec_𝕜 : DecidableEq 𝕜]
+variable {𝕜 : Type _} [RCLike 𝕜] [dec_𝕜 : DecidableEq 𝕜]
 
 variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
@@ -250,7 +250,7 @@ for a self-adjoint operator `T` on `E`.
 
 TODO Postcompose with a permutation so that these eigenvalues are listed in increasing order. -/
 noncomputable irreducible_def eigenvalues (i : Fin n) : ℝ :=
-  @IsROrC.re 𝕜 _ <|
+  @RCLike.re 𝕜 _ <|
     hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i hT.orthogonalFamily_eigenspaces'
 #align linear_map.is_symmetric.eigenvalues LinearMap.IsSymmetric.eigenvalues
 -/
@@ -264,7 +264,7 @@ theorem hasEigenvector_eigenvectorBasis (i : Fin n) :
     hT.direct_sum_is_internal.subordinate_orthonormal_basis_index hn i
       hT.orthogonal_family_eigenspaces'
   simp_rw [eigenvalues]
-  change has_eigenvector T (IsROrC.re μ) v
+  change has_eigenvector T (RCLike.re μ) v
   have key : has_eigenvector T μ v :=
     by
     have H₁ : v ∈ eigenspace T μ := by
@@ -274,8 +274,8 @@ theorem hasEigenvector_eigenvectorBasis (i : Fin n) :
           hT.orthogonal_family_eigenspaces'
     have H₂ : v ≠ 0 := by simpa using (hT.eigenvector_basis hn).toBasis.NeZero i
     exact ⟨H₁, H₂⟩
-  have re_μ : ↑(IsROrC.re μ) = μ := by
-    rw [← IsROrC.conj_eq_iff_re]
+  have re_μ : ↑(RCLike.re μ) = μ := by
+    rw [← RCLike.conj_eq_iff_re]
     exact hT.conj_eigenvalue_eq_self (has_eigenvalue_of_has_eigenvector key)
   simpa [re_μ] using key
 #align linear_map.is_symmetric.has_eigenvector_eigenvector_basis LinearMap.IsSymmetric.hasEigenvector_eigenvectorBasis
@@ -337,24 +337,24 @@ theorem inner_product_apply_eigenvector {μ : 𝕜} {v : E} {T : E →ₗ[𝕜]
 
 #print eigenvalue_nonneg_of_nonneg /-
 theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
-    (hnn : ∀ x : E, 0 ≤ IsROrC.re ⟪x, T x⟫) : 0 ≤ μ :=
+    (hnn : ∀ x : E, 0 ≤ RCLike.re ⟪x, T x⟫) : 0 ≤ μ :=
   by
   obtain ⟨v, hv⟩ := hμ.exists_has_eigenvector
   have hpos : 0 < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2
-  have : IsROrC.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
-    exact_mod_cast congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
+  have : RCLike.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
+    exact_mod_cast congr_arg RCLike.re (inner_product_apply_eigenvector hv.1)
   exact (mul_nonneg_iff_of_pos_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_nonneg_of_nonneg eigenvalue_nonneg_of_nonneg
 -/
 
 #print eigenvalue_pos_of_pos /-
 theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
-    (hnn : ∀ x : E, 0 < IsROrC.re ⟪x, T x⟫) : 0 < μ :=
+    (hnn : ∀ x : E, 0 < RCLike.re ⟪x, T x⟫) : 0 < μ :=
   by
   obtain ⟨v, hv⟩ := hμ.exists_has_eigenvector
   have hpos : 0 < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2
-  have : IsROrC.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
-    exact_mod_cast congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
+  have : RCLike.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
+    exact_mod_cast congr_arg RCLike.re (inner_product_apply_eigenvector hv.1)
   exact (mul_pos_iff_of_pos_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_pos_of_pos eigenvalue_pos_of_pos
 -/

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

@@ -72,7 +72,7 @@ variable {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric)
 theorem invariant_orthogonalComplement_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigenspace T μ)ᗮ) :
     T v ∈ (eigenspace T μ)ᗮ := by
   intro w hw
-  have : T w = (μ : 𝕜) • w := by rwa [mem_eigenspace_iff] at hw 
+  have : T w = (μ : 𝕜) • w := by rwa [mem_eigenspace_iff] at hw
   simp [← hT w, this, inner_smul_left, hv w hw]
 #align linear_map.is_symmetric.invariant_orthogonal_eigenspace LinearMap.IsSymmetric.invariant_orthogonalComplement_eigenspace
 -/
@@ -82,7 +82,7 @@ theorem invariant_orthogonalComplement_eigenspace (μ : 𝕜) (v : E) (hv : v 
 theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ = μ :=
   by
   obtain ⟨v, hv₁, hv₂⟩ := hμ.exists_has_eigenvector
-  rw [mem_eigenspace_iff] at hv₁ 
+  rw [mem_eigenspace_iff] at hv₁
   simpa [hv₂, inner_smul_left, inner_smul_right, hv₁] using hT v v
 #align linear_map.is_symmetric.conj_eigenvalue_eq_self LinearMap.IsSymmetric.conj_eigenvalue_eq_self
 -/
@@ -96,7 +96,7 @@ theorem orthogonalFamily_eigenspaces :
   by_cases hv' : v = 0
   · simp [hv']
   have H := hT.conj_eigenvalue_eq_self (has_eigenvalue_of_has_eigenvector ⟨hv, hv'⟩)
-  rw [mem_eigenspace_iff] at hv hw 
+  rw [mem_eigenspace_iff] at hv hw
   refine' Or.resolve_left _ hμν.symm
   simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
 #align linear_map.is_symmetric.orthogonal_family_eigenspaces LinearMap.IsSymmetric.orthogonalFamily_eigenspaces

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

@@ -343,7 +343,7 @@ theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : Has
   have hpos : 0 < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2
   have : IsROrC.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
     exact_mod_cast congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
-  exact (zero_le_mul_right hpos).mp (this ▸ hnn v)
+  exact (mul_nonneg_iff_of_pos_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_nonneg_of_nonneg eigenvalue_nonneg_of_nonneg
 -/
 
@@ -355,7 +355,7 @@ theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenv
   have hpos : 0 < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2
   have : IsROrC.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
     exact_mod_cast congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
-  exact (zero_lt_mul_right hpos).mp (this ▸ hnn v)
+  exact (mul_pos_iff_of_pos_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_pos_of_pos eigenvalue_pos_of_pos
 -/
 

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

@@ -312,8 +312,7 @@ theorem eigenvectorBasis_apply_self_apply (v : E) (i : Fin n) :
       congr_arg (fun v => (hT.eigenvector_basis hn).repr v i)
         (this ((hT.eigenvector_basis hn).repr v))
   intro w
-  simp_rw [← OrthonormalBasis.sum_repr_symm, LinearMap.map_sum, LinearMap.map_smul,
-    apply_eigenvector_basis]
+  simp_rw [← OrthonormalBasis.sum_repr_symm, map_sum, LinearMap.map_smul, apply_eigenvector_basis]
   apply Fintype.sum_congr
   intro a
   rw [smul_smul, mul_comm]

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

@@ -3,10 +3,10 @@ Copyright (c) 2021 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 -/
-import Mathbin.Analysis.InnerProductSpace.Rayleigh
-import Mathbin.Analysis.InnerProductSpace.PiL2
-import Mathbin.Algebra.DirectSum.Decomposition
-import Mathbin.LinearAlgebra.Eigenspace.Minpoly
+import Analysis.InnerProductSpace.Rayleigh
+import Analysis.InnerProductSpace.PiL2
+import Algebra.DirectSum.Decomposition
+import LinearAlgebra.Eigenspace.Minpoly
 
 #align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"cff8231f04dfa33fd8f2f45792eebd862ef30cad"
 

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

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
-
-! This file was ported from Lean 3 source module analysis.inner_product_space.spectrum
-! leanprover-community/mathlib commit cff8231f04dfa33fd8f2f45792eebd862ef30cad
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.InnerProductSpace.Rayleigh
 import Mathbin.Analysis.InnerProductSpace.PiL2
 import Mathbin.Algebra.DirectSum.Decomposition
 import Mathbin.LinearAlgebra.Eigenspace.Minpoly
 
+#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"cff8231f04dfa33fd8f2f45792eebd862ef30cad"
+
 /-! # Spectral theory of self-adjoint operators
 
 > THIS FILE IS SYNCHRONIZED WITH MATHLIB4.

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

@@ -70,14 +70,17 @@ namespace IsSymmetric
 
 variable {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric)
 
+#print LinearMap.IsSymmetric.invariant_orthogonalComplement_eigenspace /-
 /-- A self-adjoint operator preserves orthogonal complements of its eigenspaces. -/
-theorem invariant_orthogonal_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigenspace T μ)ᗮ) :
+theorem invariant_orthogonalComplement_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigenspace T μ)ᗮ) :
     T v ∈ (eigenspace T μ)ᗮ := by
   intro w hw
   have : T w = (μ : 𝕜) • w := by rwa [mem_eigenspace_iff] at hw 
   simp [← hT w, this, inner_smul_left, hv w hw]
-#align linear_map.is_symmetric.invariant_orthogonal_eigenspace LinearMap.IsSymmetric.invariant_orthogonal_eigenspace
+#align linear_map.is_symmetric.invariant_orthogonal_eigenspace LinearMap.IsSymmetric.invariant_orthogonalComplement_eigenspace
+-/
 
+#print LinearMap.IsSymmetric.conj_eigenvalue_eq_self /-
 /-- The eigenvalues of a self-adjoint operator are real. -/
 theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ = μ :=
   by
@@ -85,7 +88,9 @@ theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ
   rw [mem_eigenspace_iff] at hv₁ 
   simpa [hv₂, inner_smul_left, inner_smul_right, hv₁] using hT v v
 #align linear_map.is_symmetric.conj_eigenvalue_eq_self LinearMap.IsSymmetric.conj_eigenvalue_eq_self
+-/
 
+#print LinearMap.IsSymmetric.orthogonalFamily_eigenspaces /-
 /-- The eigenspaces of a self-adjoint operator are mutually orthogonal. -/
 theorem orthogonalFamily_eigenspaces :
     OrthogonalFamily 𝕜 (fun μ => eigenspace T μ) fun μ => (eigenspace T μ).subtypeₗᵢ :=
@@ -98,53 +103,66 @@ theorem orthogonalFamily_eigenspaces :
   refine' Or.resolve_left _ hμν.symm
   simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
 #align linear_map.is_symmetric.orthogonal_family_eigenspaces LinearMap.IsSymmetric.orthogonalFamily_eigenspaces
+-/
 
+#print LinearMap.IsSymmetric.orthogonalFamily_eigenspaces' /-
 theorem orthogonalFamily_eigenspaces' :
     OrthogonalFamily 𝕜 (fun μ : Eigenvalues T => eigenspace T μ) fun μ =>
       (eigenspace T μ).subtypeₗᵢ :=
   hT.orthogonalFamily_eigenspaces.comp Subtype.coe_injective
 #align linear_map.is_symmetric.orthogonal_family_eigenspaces' LinearMap.IsSymmetric.orthogonalFamily_eigenspaces'
+-/
 
+#print LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_invariant /-
 /-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on an inner
 product space is an invariant subspace of the operator. -/
-theorem orthogonal_iSup_eigenspaces_invariant ⦃v : E⦄ (hv : v ∈ (⨆ μ, eigenspace T μ)ᗮ) :
+theorem orthogonalComplement_iSup_eigenspaces_invariant ⦃v : E⦄ (hv : v ∈ (⨆ μ, eigenspace T μ)ᗮ) :
     T v ∈ (⨆ μ, eigenspace T μ)ᗮ :=
   by
   rw [← Submodule.iInf_orthogonal] at hv ⊢
   exact T.infi_invariant hT.invariant_orthogonal_eigenspace v hv
-#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_invariant LinearMap.IsSymmetric.orthogonal_iSup_eigenspaces_invariant
+#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_invariant LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_invariant
+-/
 
+#print LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces /-
 /-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on an inner
 product space has no eigenvalues. -/
-theorem orthogonal_iSup_eigenspaces (μ : 𝕜) :
-    eigenspace (T.restrict hT.orthogonal_iSup_eigenspaces_invariant) μ = ⊥ :=
+theorem orthogonalComplement_iSup_eigenspaces (μ : 𝕜) :
+    eigenspace (T.restrict hT.orthogonalComplement_iSup_eigenspaces_invariant) μ = ⊥ :=
   by
   set p : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ
   refine' eigenspace_restrict_eq_bot hT.orthogonal_supr_eigenspaces_invariant _
   have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrtho_orthogonal_right _).mono_left (le_iSup _ _)
   exact H₂.disjoint
-#align linear_map.is_symmetric.orthogonal_supr_eigenspaces LinearMap.IsSymmetric.orthogonal_iSup_eigenspaces
+#align linear_map.is_symmetric.orthogonal_supr_eigenspaces LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces
+-/
 
 /-! ### Finite-dimensional theory -/
 
 
 variable [FiniteDimensional 𝕜 E]
 
+#print LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot /-
 /-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on a
 finite-dimensional inner product space is trivial. -/
-theorem orthogonal_iSup_eigenspaces_eq_bot : (⨆ μ, eigenspace T μ)ᗮ = ⊥ :=
+theorem orthogonalComplement_iSup_eigenspaces_eq_bot : (⨆ μ, eigenspace T μ)ᗮ = ⊥ :=
   by
   have hT' : is_symmetric _ := hT.restrict_invariant hT.orthogonal_supr_eigenspaces_invariant
   -- a self-adjoint operator on a nontrivial inner product space has an eigenvalue
   haveI := hT'.subsingleton_of_no_eigenvalue_finite_dimensional hT.orthogonal_supr_eigenspaces
   exact Submodule.eq_bot_of_subsingleton _
-#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot LinearMap.IsSymmetric.orthogonal_iSup_eigenspaces_eq_bot
+#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot
+-/
 
-theorem orthogonal_iSup_eigenspaces_eq_bot' : (⨆ μ : Eigenvalues T, eigenspace T μ)ᗮ = ⊥ :=
+#print LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot' /-
+theorem orthogonalComplement_iSup_eigenspaces_eq_bot' :
+    (⨆ μ : Eigenvalues T, eigenspace T μ)ᗮ = ⊥ :=
   show (⨆ μ : { μ // eigenspace T μ ≠ ⊥ }, eigenspace T μ)ᗮ = ⊥ by
     rw [iSup_ne_bot_subtype, hT.orthogonal_supr_eigenspaces_eq_bot]
-#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot' LinearMap.IsSymmetric.orthogonal_iSup_eigenspaces_eq_bot'
+#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot' LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot'
+-/
 
+#print LinearMap.IsSymmetric.directSumDecomposition /-
 /-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` gives
 an internal direct sum decomposition of `E`.
 
@@ -155,34 +173,45 @@ noncomputable instance directSumDecomposition [hT : Fact T.IsSymmetric] :
   hT.out.orthogonal_family_eigenspaces'.decomposition
     (submodule.orthogonal_eq_bot_iff.mp hT.out.orthogonal_supr_eigenspaces_eq_bot')
 #align linear_map.is_symmetric.direct_sum_decomposition LinearMap.IsSymmetric.directSumDecomposition
+-/
 
+#print LinearMap.IsSymmetric.directSum_decompose_apply /-
 theorem directSum_decompose_apply [hT : Fact T.IsSymmetric] (x : E) (μ : Eigenvalues T) :
     DirectSum.decompose (fun μ : Eigenvalues T => eigenspace T μ) x μ =
       orthogonalProjection (eigenspace T μ) x :=
   rfl
 #align linear_map.is_symmetric.direct_sum_decompose_apply LinearMap.IsSymmetric.directSum_decompose_apply
+-/
 
+#print LinearMap.IsSymmetric.direct_sum_isInternal /-
 /-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` gives
 an internal direct sum decomposition of `E`. -/
 theorem direct_sum_isInternal : DirectSum.IsInternal fun μ : Eigenvalues T => eigenspace T μ :=
-  hT.orthogonalFamily_eigenspaces'.isInternal_iff.mpr hT.orthogonal_iSup_eigenspaces_eq_bot'
+  hT.orthogonalFamily_eigenspaces'.isInternal_iff.mpr
+    hT.orthogonalComplement_iSup_eigenspaces_eq_bot'
 #align linear_map.is_symmetric.direct_sum_is_internal LinearMap.IsSymmetric.direct_sum_isInternal
+-/
 
 section Version1
 
+#print LinearMap.IsSymmetric.diagonalization /-
 /-- Isometry from an inner product space `E` to the direct sum of the eigenspaces of some
 self-adjoint operator `T` on `E`. -/
 noncomputable def diagonalization : E ≃ₗᵢ[𝕜] PiLp 2 fun μ : Eigenvalues T => eigenspace T μ :=
   hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily hT.orthogonalFamily_eigenspaces'
 #align linear_map.is_symmetric.diagonalization LinearMap.IsSymmetric.diagonalization
+-/
 
+#print LinearMap.IsSymmetric.diagonalization_symm_apply /-
 @[simp]
 theorem diagonalization_symm_apply (w : PiLp 2 fun μ : Eigenvalues T => eigenspace T μ) :
     hT.diagonalization.symm w = ∑ μ, w μ :=
   hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily_symm_apply hT.orthogonalFamily_eigenspaces'
     w
 #align linear_map.is_symmetric.diagonalization_symm_apply LinearMap.IsSymmetric.diagonalization_symm_apply
+-/
 
+#print LinearMap.IsSymmetric.diagonalization_apply_self_apply /-
 /-- *Diagonalization theorem*, *spectral theorem*; version 1: A self-adjoint operator `T` on a
 finite-dimensional inner product space `E` acts diagonally on the decomposition of `E` into the
 direct sum of the eigenspaces of `T`. -/
@@ -199,6 +228,7 @@ theorem diagonalization_apply_self_apply (v : E) (μ : Eigenvalues T) :
   have hwT : ∀ μ, T (w μ) = (μ : 𝕜) • w μ := fun μ => mem_eigenspace_iff.1 (w μ).2
   simp only [hwT, diagonalization_symm_apply, map_sum, Submodule.coe_smul_of_tower]
 #align linear_map.is_symmetric.diagonalization_apply_self_apply LinearMap.IsSymmetric.diagonalization_apply_self_apply
+-/
 
 end Version1
 
@@ -206,6 +236,7 @@ section Version2
 
 variable {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n)
 
+#print LinearMap.IsSymmetric.eigenvectorBasis /-
 /-- A choice of orthonormal basis of eigenvectors for self-adjoint operator `T` on a
 finite-dimensional inner product space `E`.
 
@@ -214,7 +245,9 @@ eigenvalue. -/
 noncomputable irreducible_def eigenvectorBasis : OrthonormalBasis (Fin n) 𝕜 E :=
   hT.direct_sum_isInternal.subordinateOrthonormalBasis hn hT.orthogonalFamily_eigenspaces'
 #align linear_map.is_symmetric.eigenvector_basis LinearMap.IsSymmetric.eigenvectorBasis
+-/
 
+#print LinearMap.IsSymmetric.eigenvalues /-
 /-- The sequence of real eigenvalues associated to the standard orthonormal basis of eigenvectors
 for a self-adjoint operator `T` on `E`.
 
@@ -223,7 +256,9 @@ noncomputable irreducible_def eigenvalues (i : Fin n) : ℝ :=
   @IsROrC.re 𝕜 _ <|
     hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i hT.orthogonalFamily_eigenspaces'
 #align linear_map.is_symmetric.eigenvalues LinearMap.IsSymmetric.eigenvalues
+-/
 
+#print LinearMap.IsSymmetric.hasEigenvector_eigenvectorBasis /-
 theorem hasEigenvector_eigenvectorBasis (i : Fin n) :
     HasEigenvector T (hT.Eigenvalues hn i) (hT.eigenvectorBasis hn i) :=
   by
@@ -247,21 +282,27 @@ theorem hasEigenvector_eigenvectorBasis (i : Fin n) :
     exact hT.conj_eigenvalue_eq_self (has_eigenvalue_of_has_eigenvector key)
   simpa [re_μ] using key
 #align linear_map.is_symmetric.has_eigenvector_eigenvector_basis LinearMap.IsSymmetric.hasEigenvector_eigenvectorBasis
+-/
 
+#print LinearMap.IsSymmetric.hasEigenvalue_eigenvalues /-
 theorem hasEigenvalue_eigenvalues (i : Fin n) : HasEigenvalue T (hT.Eigenvalues hn i) :=
   Module.End.hasEigenvalue_of_hasEigenvector (hT.hasEigenvector_eigenvectorBasis hn i)
 #align linear_map.is_symmetric.has_eigenvalue_eigenvalues LinearMap.IsSymmetric.hasEigenvalue_eigenvalues
+-/
 
+#print LinearMap.IsSymmetric.apply_eigenvectorBasis /-
 @[simp]
 theorem apply_eigenvectorBasis (i : Fin n) :
     T (hT.eigenvectorBasis hn i) = (hT.Eigenvalues hn i : 𝕜) • hT.eigenvectorBasis hn i :=
   mem_eigenspace_iff.mp (hT.hasEigenvector_eigenvectorBasis hn i).1
 #align linear_map.is_symmetric.apply_eigenvector_basis LinearMap.IsSymmetric.apply_eigenvectorBasis
+-/
 
+#print LinearMap.IsSymmetric.eigenvectorBasis_apply_self_apply /-
 /-- *Diagonalization theorem*, *spectral theorem*; version 2: A self-adjoint operator `T` on a
 finite-dimensional inner product space `E` acts diagonally on the identification of `E` with
 Euclidean space induced by an orthonormal basis of eigenvectors of `T`. -/
-theorem diagonalization_basis_apply_self_apply (v : E) (i : Fin n) :
+theorem eigenvectorBasis_apply_self_apply (v : E) (i : Fin n) :
     (hT.eigenvectorBasis hn).repr (T v) i =
       hT.Eigenvalues hn i * (hT.eigenvectorBasis hn).repr v i :=
   by
@@ -279,7 +320,8 @@ theorem diagonalization_basis_apply_self_apply (v : E) (i : Fin n) :
   apply Fintype.sum_congr
   intro a
   rw [smul_smul, mul_comm]
-#align linear_map.is_symmetric.diagonalization_basis_apply_self_apply LinearMap.IsSymmetric.diagonalization_basis_apply_self_apply
+#align linear_map.is_symmetric.diagonalization_basis_apply_self_apply LinearMap.IsSymmetric.eigenvectorBasis_apply_self_apply
+-/
 
 end Version2
 
@@ -289,12 +331,15 @@ end LinearMap
 
 section Nonneg
 
+#print inner_product_apply_eigenvector /-
 @[simp]
 theorem inner_product_apply_eigenvector {μ : 𝕜} {v : E} {T : E →ₗ[𝕜] E}
     (h : v ∈ Module.End.eigenspace T μ) : ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
   simp only [mem_eigenspace_iff.mp h, inner_smul_right, inner_self_eq_norm_sq_to_K]
 #align inner_product_apply_eigenvector inner_product_apply_eigenvector
+-/
 
+#print eigenvalue_nonneg_of_nonneg /-
 theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
     (hnn : ∀ x : E, 0 ≤ IsROrC.re ⟪x, T x⟫) : 0 ≤ μ :=
   by
@@ -304,7 +349,9 @@ theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : Has
     exact_mod_cast congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
   exact (zero_le_mul_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_nonneg_of_nonneg eigenvalue_nonneg_of_nonneg
+-/
 
+#print eigenvalue_pos_of_pos /-
 theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
     (hnn : ∀ x : E, 0 < IsROrC.re ⟪x, T x⟫) : 0 < μ :=
   by
@@ -314,6 +361,7 @@ theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenv
     exact_mod_cast congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
   exact (zero_lt_mul_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_pos_of_pos eigenvalue_pos_of_pos
+-/
 
 end Nonneg
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/893964fc28cefbcffc7cb784ed00a2895b4e65cf

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.spectrum
-! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
+! leanprover-community/mathlib commit cff8231f04dfa33fd8f2f45792eebd862ef30cad
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.LinearAlgebra.Eigenspace.Minpoly
 
 /-! # Spectral theory of self-adjoint operators
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file covers the spectral theory of self-adjoint operators on an inner product space.
 
 The first part of the file covers general properties, true without any condition on boundedness or

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

@@ -55,7 +55,6 @@ variable {𝕜 : Type _} [IsROrC 𝕜] [dec_𝕜 : DecidableEq 𝕜]
 
 variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
--- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 E _ x y
 
 open scoped BigOperators ComplexConjugate
@@ -68,8 +67,6 @@ namespace IsSymmetric
 
 variable {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric)
 
-include hT
-
 /-- A self-adjoint operator preserves orthogonal complements of its eigenspaces. -/
 theorem invariant_orthogonal_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigenspace T μ)ᗮ) :
     T v ∈ (eigenspace T μ)ᗮ := by
@@ -145,10 +142,6 @@ theorem orthogonal_iSup_eigenspaces_eq_bot' : (⨆ μ : Eigenvalues T, eigenspac
     rw [iSup_ne_bot_subtype, hT.orthogonal_supr_eigenspaces_eq_bot]
 #align linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot' LinearMap.IsSymmetric.orthogonal_iSup_eigenspaces_eq_bot'
 
-include dec_𝕜
-
-omit hT
-
 /-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` gives
 an internal direct sum decomposition of `E`.
 
@@ -166,8 +159,6 @@ theorem directSum_decompose_apply [hT : Fact T.IsSymmetric] (x : E) (μ : Eigenv
   rfl
 #align linear_map.is_symmetric.direct_sum_decompose_apply LinearMap.IsSymmetric.directSum_decompose_apply
 
-include hT
-
 /-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` gives
 an internal direct sum decomposition of `E`. -/
 theorem direct_sum_isInternal : DirectSum.IsInternal fun μ : Eigenvalues T => eigenspace T μ :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -4,13 +4,14 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.spectrum
-! leanprover-community/mathlib commit a33d01f7f8d39883e1ab4b86c9bf21692f1d4356
+! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.InnerProductSpace.Rayleigh
 import Mathbin.Analysis.InnerProductSpace.PiL2
 import Mathbin.Algebra.DirectSum.Decomposition
+import Mathbin.LinearAlgebra.Eigenspace.Minpoly
 
 /-! # Spectral theory of self-adjoint operators
 

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

@@ -73,7 +73,7 @@ include hT
 theorem invariant_orthogonal_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigenspace T μ)ᗮ) :
     T v ∈ (eigenspace T μ)ᗮ := by
   intro w hw
-  have : T w = (μ : 𝕜) • w := by rwa [mem_eigenspace_iff] at hw
+  have : T w = (μ : 𝕜) • w := by rwa [mem_eigenspace_iff] at hw 
   simp [← hT w, this, inner_smul_left, hv w hw]
 #align linear_map.is_symmetric.invariant_orthogonal_eigenspace LinearMap.IsSymmetric.invariant_orthogonal_eigenspace
 
@@ -81,7 +81,7 @@ theorem invariant_orthogonal_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigensp
 theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ = μ :=
   by
   obtain ⟨v, hv₁, hv₂⟩ := hμ.exists_has_eigenvector
-  rw [mem_eigenspace_iff] at hv₁
+  rw [mem_eigenspace_iff] at hv₁ 
   simpa [hv₂, inner_smul_left, inner_smul_right, hv₁] using hT v v
 #align linear_map.is_symmetric.conj_eigenvalue_eq_self LinearMap.IsSymmetric.conj_eigenvalue_eq_self
 
@@ -93,7 +93,7 @@ theorem orthogonalFamily_eigenspaces :
   by_cases hv' : v = 0
   · simp [hv']
   have H := hT.conj_eigenvalue_eq_self (has_eigenvalue_of_has_eigenvector ⟨hv, hv'⟩)
-  rw [mem_eigenspace_iff] at hv hw
+  rw [mem_eigenspace_iff] at hv hw 
   refine' Or.resolve_left _ hμν.symm
   simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
 #align linear_map.is_symmetric.orthogonal_family_eigenspaces LinearMap.IsSymmetric.orthogonalFamily_eigenspaces
@@ -109,7 +109,7 @@ product space is an invariant subspace of the operator. -/
 theorem orthogonal_iSup_eigenspaces_invariant ⦃v : E⦄ (hv : v ∈ (⨆ μ, eigenspace T μ)ᗮ) :
     T v ∈ (⨆ μ, eigenspace T μ)ᗮ :=
   by
-  rw [← Submodule.iInf_orthogonal] at hv⊢
+  rw [← Submodule.iInf_orthogonal] at hv ⊢
   exact T.infi_invariant hT.invariant_orthogonal_eigenspace v hv
 #align linear_map.is_symmetric.orthogonal_supr_eigenspaces_invariant LinearMap.IsSymmetric.orthogonal_iSup_eigenspaces_invariant
 

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

@@ -57,7 +57,7 @@ variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 -- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 E _ x y
 
-open BigOperators ComplexConjugate
+open scoped BigOperators ComplexConjugate
 
 open Module.End
 

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

@@ -4,12 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.spectrum
-! leanprover-community/mathlib commit caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c
+! leanprover-community/mathlib commit a33d01f7f8d39883e1ab4b86c9bf21692f1d4356
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.InnerProductSpace.Rayleigh
 import Mathbin.Analysis.InnerProductSpace.PiL2
+import Mathbin.Algebra.DirectSum.Decomposition
 
 /-! # Spectral theory of self-adjoint operators
 
@@ -145,7 +146,28 @@ theorem orthogonal_iSup_eigenspaces_eq_bot' : (⨆ μ : Eigenvalues T, eigenspac
 
 include dec_𝕜
 
-/-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` give
+omit hT
+
+/-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` gives
+an internal direct sum decomposition of `E`.
+
+Note this takes `hT` as a `fact` to allow it to be an instance. -/
+noncomputable instance directSumDecomposition [hT : Fact T.IsSymmetric] :
+    DirectSum.Decomposition fun μ : Eigenvalues T => eigenspace T μ :=
+  haveI h : ∀ μ : eigenvalues T, CompleteSpace (eigenspace T μ) := fun μ => by infer_instance
+  hT.out.orthogonal_family_eigenspaces'.decomposition
+    (submodule.orthogonal_eq_bot_iff.mp hT.out.orthogonal_supr_eigenspaces_eq_bot')
+#align linear_map.is_symmetric.direct_sum_decomposition LinearMap.IsSymmetric.directSumDecomposition
+
+theorem directSum_decompose_apply [hT : Fact T.IsSymmetric] (x : E) (μ : Eigenvalues T) :
+    DirectSum.decompose (fun μ : Eigenvalues T => eigenspace T μ) x μ =
+      orthogonalProjection (eigenspace T μ) x :=
+  rfl
+#align linear_map.is_symmetric.direct_sum_decompose_apply LinearMap.IsSymmetric.directSum_decompose_apply
+
+include hT
+
+/-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` gives
 an internal direct sum decomposition of `E`. -/
 theorem direct_sum_isInternal : DirectSum.IsInternal fun μ : Eigenvalues T => eigenspace T μ :=
   hT.orthogonalFamily_eigenspaces'.isInternal_iff.mpr hT.orthogonal_iSup_eigenspaces_eq_bot'

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

@@ -85,7 +85,7 @@ theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ
 #align linear_map.is_symmetric.conj_eigenvalue_eq_self LinearMap.IsSymmetric.conj_eigenvalue_eq_self
 
 /-- The eigenspaces of a self-adjoint operator are mutually orthogonal. -/
-theorem orthogonalFamilyEigenspaces :
+theorem orthogonalFamily_eigenspaces :
     OrthogonalFamily 𝕜 (fun μ => eigenspace T μ) fun μ => (eigenspace T μ).subtypeₗᵢ :=
   by
   rintro μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩
@@ -95,13 +95,13 @@ theorem orthogonalFamilyEigenspaces :
   rw [mem_eigenspace_iff] at hv hw
   refine' Or.resolve_left _ hμν.symm
   simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
-#align linear_map.is_symmetric.orthogonal_family_eigenspaces LinearMap.IsSymmetric.orthogonalFamilyEigenspaces
+#align linear_map.is_symmetric.orthogonal_family_eigenspaces LinearMap.IsSymmetric.orthogonalFamily_eigenspaces
 
-theorem orthogonalFamilyEigenspaces' :
+theorem orthogonalFamily_eigenspaces' :
     OrthogonalFamily 𝕜 (fun μ : Eigenvalues T => eigenspace T μ) fun μ =>
       (eigenspace T μ).subtypeₗᵢ :=
-  hT.orthogonalFamilyEigenspaces.comp Subtype.coe_injective
-#align linear_map.is_symmetric.orthogonal_family_eigenspaces' LinearMap.IsSymmetric.orthogonalFamilyEigenspaces'
+  hT.orthogonalFamily_eigenspaces.comp Subtype.coe_injective
+#align linear_map.is_symmetric.orthogonal_family_eigenspaces' LinearMap.IsSymmetric.orthogonalFamily_eigenspaces'
 
 /-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on an inner
 product space is an invariant subspace of the operator. -/
@@ -119,7 +119,7 @@ theorem orthogonal_iSup_eigenspaces (μ : 𝕜) :
   by
   set p : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ
   refine' eigenspace_restrict_eq_bot hT.orthogonal_supr_eigenspaces_invariant _
-  have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrthoOrthogonalRight _).mono_left (le_iSup _ _)
+  have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrtho_orthogonal_right _).mono_left (le_iSup _ _)
   exact H₂.disjoint
 #align linear_map.is_symmetric.orthogonal_supr_eigenspaces LinearMap.IsSymmetric.orthogonal_iSup_eigenspaces
 
@@ -148,7 +148,7 @@ include dec_𝕜
 /-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` give
 an internal direct sum decomposition of `E`. -/
 theorem direct_sum_isInternal : DirectSum.IsInternal fun μ : Eigenvalues T => eigenspace T μ :=
-  hT.orthogonalFamilyEigenspaces'.isInternal_iff.mpr hT.orthogonal_iSup_eigenspaces_eq_bot'
+  hT.orthogonalFamily_eigenspaces'.isInternal_iff.mpr hT.orthogonal_iSup_eigenspaces_eq_bot'
 #align linear_map.is_symmetric.direct_sum_is_internal LinearMap.IsSymmetric.direct_sum_isInternal
 
 section Version1
@@ -156,13 +156,14 @@ section Version1
 /-- Isometry from an inner product space `E` to the direct sum of the eigenspaces of some
 self-adjoint operator `T` on `E`. -/
 noncomputable def diagonalization : E ≃ₗᵢ[𝕜] PiLp 2 fun μ : Eigenvalues T => eigenspace T μ :=
-  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily hT.orthogonalFamilyEigenspaces'
+  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily hT.orthogonalFamily_eigenspaces'
 #align linear_map.is_symmetric.diagonalization LinearMap.IsSymmetric.diagonalization
 
 @[simp]
 theorem diagonalization_symm_apply (w : PiLp 2 fun μ : Eigenvalues T => eigenspace T μ) :
     hT.diagonalization.symm w = ∑ μ, w μ :=
-  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily_symm_apply hT.orthogonalFamilyEigenspaces' w
+  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily_symm_apply hT.orthogonalFamily_eigenspaces'
+    w
 #align linear_map.is_symmetric.diagonalization_symm_apply LinearMap.IsSymmetric.diagonalization_symm_apply
 
 /-- *Diagonalization theorem*, *spectral theorem*; version 1: A self-adjoint operator `T` on a
@@ -194,7 +195,7 @@ finite-dimensional inner product space `E`.
 TODO Postcompose with a permutation so that these eigenvectors are listed in increasing order of
 eigenvalue. -/
 noncomputable irreducible_def eigenvectorBasis : OrthonormalBasis (Fin n) 𝕜 E :=
-  hT.direct_sum_isInternal.subordinateOrthonormalBasis hn hT.orthogonalFamilyEigenspaces'
+  hT.direct_sum_isInternal.subordinateOrthonormalBasis hn hT.orthogonalFamily_eigenspaces'
 #align linear_map.is_symmetric.eigenvector_basis LinearMap.IsSymmetric.eigenvectorBasis
 
 /-- The sequence of real eigenvalues associated to the standard orthonormal basis of eigenvectors
@@ -203,7 +204,7 @@ for a self-adjoint operator `T` on `E`.
 TODO Postcompose with a permutation so that these eigenvalues are listed in increasing order. -/
 noncomputable irreducible_def eigenvalues (i : Fin n) : ℝ :=
   @IsROrC.re 𝕜 _ <|
-    hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i hT.orthogonalFamilyEigenspaces'
+    hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i hT.orthogonalFamily_eigenspaces'
 #align linear_map.is_symmetric.eigenvalues LinearMap.IsSymmetric.eigenvalues
 
 theorem hasEigenvector_eigenvectorBasis (i : Fin n) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -105,23 +105,23 @@ theorem orthogonalFamilyEigenspaces' :
 
 /-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on an inner
 product space is an invariant subspace of the operator. -/
-theorem orthogonal_supᵢ_eigenspaces_invariant ⦃v : E⦄ (hv : v ∈ (⨆ μ, eigenspace T μ)ᗮ) :
+theorem orthogonal_iSup_eigenspaces_invariant ⦃v : E⦄ (hv : v ∈ (⨆ μ, eigenspace T μ)ᗮ) :
     T v ∈ (⨆ μ, eigenspace T μ)ᗮ :=
   by
-  rw [← Submodule.infᵢ_orthogonal] at hv⊢
+  rw [← Submodule.iInf_orthogonal] at hv⊢
   exact T.infi_invariant hT.invariant_orthogonal_eigenspace v hv
-#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_invariant LinearMap.IsSymmetric.orthogonal_supᵢ_eigenspaces_invariant
+#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_invariant LinearMap.IsSymmetric.orthogonal_iSup_eigenspaces_invariant
 
 /-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on an inner
 product space has no eigenvalues. -/
-theorem orthogonal_supᵢ_eigenspaces (μ : 𝕜) :
-    eigenspace (T.restrict hT.orthogonal_supᵢ_eigenspaces_invariant) μ = ⊥ :=
+theorem orthogonal_iSup_eigenspaces (μ : 𝕜) :
+    eigenspace (T.restrict hT.orthogonal_iSup_eigenspaces_invariant) μ = ⊥ :=
   by
   set p : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ
   refine' eigenspace_restrict_eq_bot hT.orthogonal_supr_eigenspaces_invariant _
-  have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrthoOrthogonalRight _).mono_left (le_supᵢ _ _)
+  have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrthoOrthogonalRight _).mono_left (le_iSup _ _)
   exact H₂.disjoint
-#align linear_map.is_symmetric.orthogonal_supr_eigenspaces LinearMap.IsSymmetric.orthogonal_supᵢ_eigenspaces
+#align linear_map.is_symmetric.orthogonal_supr_eigenspaces LinearMap.IsSymmetric.orthogonal_iSup_eigenspaces
 
 /-! ### Finite-dimensional theory -/
 
@@ -130,25 +130,25 @@ variable [FiniteDimensional 𝕜 E]
 
 /-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on a
 finite-dimensional inner product space is trivial. -/
-theorem orthogonal_supᵢ_eigenspaces_eq_bot : (⨆ μ, eigenspace T μ)ᗮ = ⊥ :=
+theorem orthogonal_iSup_eigenspaces_eq_bot : (⨆ μ, eigenspace T μ)ᗮ = ⊥ :=
   by
   have hT' : is_symmetric _ := hT.restrict_invariant hT.orthogonal_supr_eigenspaces_invariant
   -- a self-adjoint operator on a nontrivial inner product space has an eigenvalue
   haveI := hT'.subsingleton_of_no_eigenvalue_finite_dimensional hT.orthogonal_supr_eigenspaces
   exact Submodule.eq_bot_of_subsingleton _
-#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot LinearMap.IsSymmetric.orthogonal_supᵢ_eigenspaces_eq_bot
+#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot LinearMap.IsSymmetric.orthogonal_iSup_eigenspaces_eq_bot
 
-theorem orthogonal_supᵢ_eigenspaces_eq_bot' : (⨆ μ : Eigenvalues T, eigenspace T μ)ᗮ = ⊥ :=
+theorem orthogonal_iSup_eigenspaces_eq_bot' : (⨆ μ : Eigenvalues T, eigenspace T μ)ᗮ = ⊥ :=
   show (⨆ μ : { μ // eigenspace T μ ≠ ⊥ }, eigenspace T μ)ᗮ = ⊥ by
-    rw [supᵢ_ne_bot_subtype, hT.orthogonal_supr_eigenspaces_eq_bot]
-#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot' LinearMap.IsSymmetric.orthogonal_supᵢ_eigenspaces_eq_bot'
+    rw [iSup_ne_bot_subtype, hT.orthogonal_supr_eigenspaces_eq_bot]
+#align linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot' LinearMap.IsSymmetric.orthogonal_iSup_eigenspaces_eq_bot'
 
 include dec_𝕜
 
 /-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` give
 an internal direct sum decomposition of `E`. -/
 theorem direct_sum_isInternal : DirectSum.IsInternal fun μ : Eigenvalues T => eigenspace T μ :=
-  hT.orthogonalFamilyEigenspaces'.isInternal_iff.mpr hT.orthogonal_supᵢ_eigenspaces_eq_bot'
+  hT.orthogonalFamilyEigenspaces'.isInternal_iff.mpr hT.orthogonal_iSup_eigenspaces_eq_bot'
 #align linear_map.is_symmetric.direct_sum_is_internal LinearMap.IsSymmetric.direct_sum_isInternal
 
 section Version1

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

@@ -85,7 +85,7 @@ theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ
 #align linear_map.is_symmetric.conj_eigenvalue_eq_self LinearMap.IsSymmetric.conj_eigenvalue_eq_self
 
 /-- The eigenspaces of a self-adjoint operator are mutually orthogonal. -/
-theorem orthogonalFamily_eigenspaces :
+theorem orthogonalFamilyEigenspaces :
     OrthogonalFamily 𝕜 (fun μ => eigenspace T μ) fun μ => (eigenspace T μ).subtypeₗᵢ :=
   by
   rintro μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩
@@ -95,13 +95,13 @@ theorem orthogonalFamily_eigenspaces :
   rw [mem_eigenspace_iff] at hv hw
   refine' Or.resolve_left _ hμν.symm
   simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
-#align linear_map.is_symmetric.orthogonal_family_eigenspaces LinearMap.IsSymmetric.orthogonalFamily_eigenspaces
+#align linear_map.is_symmetric.orthogonal_family_eigenspaces LinearMap.IsSymmetric.orthogonalFamilyEigenspaces
 
-theorem orthogonalFamily_eigenspaces' :
+theorem orthogonalFamilyEigenspaces' :
     OrthogonalFamily 𝕜 (fun μ : Eigenvalues T => eigenspace T μ) fun μ =>
       (eigenspace T μ).subtypeₗᵢ :=
-  hT.orthogonalFamily_eigenspaces.comp Subtype.coe_injective
-#align linear_map.is_symmetric.orthogonal_family_eigenspaces' LinearMap.IsSymmetric.orthogonalFamily_eigenspaces'
+  hT.orthogonalFamilyEigenspaces.comp Subtype.coe_injective
+#align linear_map.is_symmetric.orthogonal_family_eigenspaces' LinearMap.IsSymmetric.orthogonalFamilyEigenspaces'
 
 /-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on an inner
 product space is an invariant subspace of the operator. -/
@@ -119,7 +119,7 @@ theorem orthogonal_supᵢ_eigenspaces (μ : 𝕜) :
   by
   set p : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ
   refine' eigenspace_restrict_eq_bot hT.orthogonal_supr_eigenspaces_invariant _
-  have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrtho_orthogonal_right _).mono_left (le_supᵢ _ _)
+  have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrthoOrthogonalRight _).mono_left (le_supᵢ _ _)
   exact H₂.disjoint
 #align linear_map.is_symmetric.orthogonal_supr_eigenspaces LinearMap.IsSymmetric.orthogonal_supᵢ_eigenspaces
 
@@ -148,7 +148,7 @@ include dec_𝕜
 /-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` give
 an internal direct sum decomposition of `E`. -/
 theorem direct_sum_isInternal : DirectSum.IsInternal fun μ : Eigenvalues T => eigenspace T μ :=
-  hT.orthogonalFamily_eigenspaces'.isInternal_iff.mpr hT.orthogonal_supᵢ_eigenspaces_eq_bot'
+  hT.orthogonalFamilyEigenspaces'.isInternal_iff.mpr hT.orthogonal_supᵢ_eigenspaces_eq_bot'
 #align linear_map.is_symmetric.direct_sum_is_internal LinearMap.IsSymmetric.direct_sum_isInternal
 
 section Version1
@@ -156,14 +156,13 @@ section Version1
 /-- Isometry from an inner product space `E` to the direct sum of the eigenspaces of some
 self-adjoint operator `T` on `E`. -/
 noncomputable def diagonalization : E ≃ₗᵢ[𝕜] PiLp 2 fun μ : Eigenvalues T => eigenspace T μ :=
-  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily hT.orthogonalFamily_eigenspaces'
+  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily hT.orthogonalFamilyEigenspaces'
 #align linear_map.is_symmetric.diagonalization LinearMap.IsSymmetric.diagonalization
 
 @[simp]
 theorem diagonalization_symm_apply (w : PiLp 2 fun μ : Eigenvalues T => eigenspace T μ) :
     hT.diagonalization.symm w = ∑ μ, w μ :=
-  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily_symm_apply hT.orthogonalFamily_eigenspaces'
-    w
+  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily_symm_apply hT.orthogonalFamilyEigenspaces' w
 #align linear_map.is_symmetric.diagonalization_symm_apply LinearMap.IsSymmetric.diagonalization_symm_apply
 
 /-- *Diagonalization theorem*, *spectral theorem*; version 1: A self-adjoint operator `T` on a
@@ -195,7 +194,7 @@ finite-dimensional inner product space `E`.
 TODO Postcompose with a permutation so that these eigenvectors are listed in increasing order of
 eigenvalue. -/
 noncomputable irreducible_def eigenvectorBasis : OrthonormalBasis (Fin n) 𝕜 E :=
-  hT.direct_sum_isInternal.subordinateOrthonormalBasis hn hT.orthogonalFamily_eigenspaces'
+  hT.direct_sum_isInternal.subordinateOrthonormalBasis hn hT.orthogonalFamilyEigenspaces'
 #align linear_map.is_symmetric.eigenvector_basis LinearMap.IsSymmetric.eigenvectorBasis
 
 /-- The sequence of real eigenvalues associated to the standard orthonormal basis of eigenvectors
@@ -204,7 +203,7 @@ for a self-adjoint operator `T` on `E`.
 TODO Postcompose with a permutation so that these eigenvalues are listed in increasing order. -/
 noncomputable irreducible_def eigenvalues (i : Fin n) : ℝ :=
   @IsROrC.re 𝕜 _ <|
-    hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i hT.orthogonalFamily_eigenspaces'
+    hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i hT.orthogonalFamilyEigenspaces'
 #align linear_map.is_symmetric.eigenvalues LinearMap.IsSymmetric.eigenvalues
 
 theorem hasEigenvector_eigenvectorBasis (i : Fin n) :

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: Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.spectrum
-! leanprover-community/mathlib commit 1a4df69ca1a9a0e5e26bfe12e2b92814216016d0
+! leanprover-community/mathlib commit caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -226,7 +226,7 @@ theorem hasEigenvector_eigenvectorBasis (i : Fin n) :
     have H₂ : v ≠ 0 := by simpa using (hT.eigenvector_basis hn).toBasis.NeZero i
     exact ⟨H₁, H₂⟩
   have re_μ : ↑(IsROrC.re μ) = μ := by
-    rw [← IsROrC.eq_conj_iff_re]
+    rw [← IsROrC.conj_eq_iff_re]
     exact hT.conj_eigenvalue_eq_self (has_eigenvalue_of_has_eigenvector key)
   simpa [re_μ] using key
 #align linear_map.is_symmetric.has_eigenvector_eigenvector_basis LinearMap.IsSymmetric.hasEigenvector_eigenvectorBasis

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a4df69ca1a9a0e5e26bfe12e2b92814216016d0

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.spectrum
-! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
+! leanprover-community/mathlib commit 1a4df69ca1a9a0e5e26bfe12e2b92814216016d0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -119,8 +119,8 @@ theorem orthogonal_supᵢ_eigenspaces (μ : 𝕜) :
   by
   set p : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ
   refine' eigenspace_restrict_eq_bot hT.orthogonal_supr_eigenspaces_invariant _
-  have H₂ : p ≤ (eigenspace T μ)ᗮ := Submodule.orthogonal_le (le_supᵢ _ _)
-  exact (eigenspace T μ).orthogonal_disjoint.mono_right H₂
+  have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrtho_orthogonal_right _).mono_left (le_supᵢ _ _)
+  exact H₂.disjoint
 #align linear_map.is_symmetric.orthogonal_supr_eigenspaces LinearMap.IsSymmetric.orthogonal_supᵢ_eigenspaces
 
 /-! ### Finite-dimensional theory -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/55d771df074d0dd020139ee1cd4b95521422df9f

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.spectrum
-! leanprover-community/mathlib commit 4681620dafca6a7d710f437bd10fb69428ec2209
+! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -51,7 +51,7 @@ self-adjoint operator, spectral theorem, diagonalization theorem
 
 variable {𝕜 : Type _} [IsROrC 𝕜] [dec_𝕜 : DecidableEq 𝕜]
 
-variable {E : Type _} [InnerProductSpace 𝕜 E]
+variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 -- mathport name: «expr⟪ , ⟫»
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 E _ x y

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.spectrum
-! leanprover-community/mathlib commit 989433cb1ffb03ed63e549ef85da37da12a5a5f6
+! leanprover-community/mathlib commit 4681620dafca6a7d710f437bd10fb69428ec2209
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -86,7 +86,7 @@ theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ
 
 /-- The eigenspaces of a self-adjoint operator are mutually orthogonal. -/
 theorem orthogonalFamily_eigenspaces :
-    @OrthogonalFamily 𝕜 _ _ _ _ (fun μ => eigenspace T μ) _ fun μ => (eigenspace T μ).subtypeₗᵢ :=
+    OrthogonalFamily 𝕜 (fun μ => eigenspace T μ) fun μ => (eigenspace T μ).subtypeₗᵢ :=
   by
   rintro μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩
   by_cases hv' : v = 0
@@ -98,7 +98,7 @@ theorem orthogonalFamily_eigenspaces :
 #align linear_map.is_symmetric.orthogonal_family_eigenspaces LinearMap.IsSymmetric.orthogonalFamily_eigenspaces
 
 theorem orthogonalFamily_eigenspaces' :
-    @OrthogonalFamily 𝕜 _ _ _ _ (fun μ : Eigenvalues T => eigenspace T μ) _ fun μ =>
+    OrthogonalFamily 𝕜 (fun μ : Eigenvalues T => eigenspace T μ) fun μ =>
       (eigenspace T μ).subtypeₗᵢ :=
   hT.orthogonalFamily_eigenspaces.comp Subtype.coe_injective
 #align linear_map.is_symmetric.orthogonal_family_eigenspaces' LinearMap.IsSymmetric.orthogonalFamily_eigenspaces'

mathlib commit https://github.com/leanprover-community/mathlib/commit/2af0836443b4cfb5feda0df0051acdb398304931

@@ -85,7 +85,7 @@ theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ
 #align linear_map.is_symmetric.conj_eigenvalue_eq_self LinearMap.IsSymmetric.conj_eigenvalue_eq_self
 
 /-- The eigenspaces of a self-adjoint operator are mutually orthogonal. -/
-theorem orthogonalFamilyEigenspaces :
+theorem orthogonalFamily_eigenspaces :
     @OrthogonalFamily 𝕜 _ _ _ _ (fun μ => eigenspace T μ) _ fun μ => (eigenspace T μ).subtypeₗᵢ :=
   by
   rintro μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩
@@ -95,13 +95,13 @@ theorem orthogonalFamilyEigenspaces :
   rw [mem_eigenspace_iff] at hv hw
   refine' Or.resolve_left _ hμν.symm
   simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
-#align linear_map.is_symmetric.orthogonal_family_eigenspaces LinearMap.IsSymmetric.orthogonalFamilyEigenspaces
+#align linear_map.is_symmetric.orthogonal_family_eigenspaces LinearMap.IsSymmetric.orthogonalFamily_eigenspaces
 
-theorem orthogonalFamilyEigenspaces' :
+theorem orthogonalFamily_eigenspaces' :
     @OrthogonalFamily 𝕜 _ _ _ _ (fun μ : Eigenvalues T => eigenspace T μ) _ fun μ =>
       (eigenspace T μ).subtypeₗᵢ :=
-  hT.orthogonalFamilyEigenspaces.comp Subtype.coe_injective
-#align linear_map.is_symmetric.orthogonal_family_eigenspaces' LinearMap.IsSymmetric.orthogonalFamilyEigenspaces'
+  hT.orthogonalFamily_eigenspaces.comp Subtype.coe_injective
+#align linear_map.is_symmetric.orthogonal_family_eigenspaces' LinearMap.IsSymmetric.orthogonalFamily_eigenspaces'
 
 /-- The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on an inner
 product space is an invariant subspace of the operator. -/
@@ -148,7 +148,7 @@ include dec_𝕜
 /-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` give
 an internal direct sum decomposition of `E`. -/
 theorem direct_sum_isInternal : DirectSum.IsInternal fun μ : Eigenvalues T => eigenspace T μ :=
-  hT.orthogonalFamilyEigenspaces'.isInternal_iff.mpr hT.orthogonal_supᵢ_eigenspaces_eq_bot'
+  hT.orthogonalFamily_eigenspaces'.isInternal_iff.mpr hT.orthogonal_supᵢ_eigenspaces_eq_bot'
 #align linear_map.is_symmetric.direct_sum_is_internal LinearMap.IsSymmetric.direct_sum_isInternal
 
 section Version1
@@ -156,13 +156,14 @@ section Version1
 /-- Isometry from an inner product space `E` to the direct sum of the eigenspaces of some
 self-adjoint operator `T` on `E`. -/
 noncomputable def diagonalization : E ≃ₗᵢ[𝕜] PiLp 2 fun μ : Eigenvalues T => eigenspace T μ :=
-  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily hT.orthogonalFamilyEigenspaces'
+  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily hT.orthogonalFamily_eigenspaces'
 #align linear_map.is_symmetric.diagonalization LinearMap.IsSymmetric.diagonalization
 
 @[simp]
 theorem diagonalization_symm_apply (w : PiLp 2 fun μ : Eigenvalues T => eigenspace T μ) :
     hT.diagonalization.symm w = ∑ μ, w μ :=
-  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily_symm_apply hT.orthogonalFamilyEigenspaces' w
+  hT.direct_sum_isInternal.isometryL2OfOrthogonalFamily_symm_apply hT.orthogonalFamily_eigenspaces'
+    w
 #align linear_map.is_symmetric.diagonalization_symm_apply LinearMap.IsSymmetric.diagonalization_symm_apply
 
 /-- *Diagonalization theorem*, *spectral theorem*; version 1: A self-adjoint operator `T` on a
@@ -194,7 +195,7 @@ finite-dimensional inner product space `E`.
 TODO Postcompose with a permutation so that these eigenvectors are listed in increasing order of
 eigenvalue. -/
 noncomputable irreducible_def eigenvectorBasis : OrthonormalBasis (Fin n) 𝕜 E :=
-  hT.direct_sum_isInternal.subordinateOrthonormalBasis hn hT.orthogonalFamilyEigenspaces'
+  hT.direct_sum_isInternal.subordinateOrthonormalBasis hn hT.orthogonalFamily_eigenspaces'
 #align linear_map.is_symmetric.eigenvector_basis LinearMap.IsSymmetric.eigenvectorBasis
 
 /-- The sequence of real eigenvalues associated to the standard orthonormal basis of eigenvectors
@@ -203,7 +204,7 @@ for a self-adjoint operator `T` on `E`.
 TODO Postcompose with a permutation so that these eigenvalues are listed in increasing order. -/
 noncomputable irreducible_def eigenvalues (i : Fin n) : ℝ :=
   @IsROrC.re 𝕜 _ <|
-    hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i hT.orthogonalFamilyEigenspaces'
+    hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i hT.orthogonalFamily_eigenspaces'
 #align linear_map.is_symmetric.eigenvalues LinearMap.IsSymmetric.eigenvalues
 
 theorem hasEigenvector_eigenvectorBasis (i : Fin n) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/195fcd60ff2bfe392543bceb0ec2adcdb472db4c

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.inner_product_space.spectrum
-! leanprover-community/mathlib commit 39afa0b340a56b158991e109cf9d1e5396e67f60
+! leanprover-community/mathlib commit 989433cb1ffb03ed63e549ef85da37da12a5a5f6
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -175,14 +175,11 @@ theorem diagonalization_apply_self_apply (v : E) (μ : Eigenvalues T) :
     ∀ w : PiLp 2 fun μ : eigenvalues T => eigenspace T μ,
       T (hT.diagonalization.symm w) = hT.diagonalization.symm fun μ => (μ : 𝕜) • w μ
     by
-    simpa [LinearIsometryEquiv.symm_apply_apply, -is_symmetric.diagonalization_symm_apply] using
+    simpa only [LinearIsometryEquiv.symm_apply_apply, LinearIsometryEquiv.apply_symm_apply] using
       congr_arg (fun w => hT.diagonalization w μ) (this (hT.diagonalization v))
   intro w
-  have hwT : ∀ μ : eigenvalues T, T (w μ) = (μ : 𝕜) • w μ :=
-    by
-    intro μ
-    simpa [mem_eigenspace_iff] using (w μ).Prop
-  simp [hwT]
+  have hwT : ∀ μ, T (w μ) = (μ : 𝕜) • w μ := fun μ => mem_eigenspace_iff.1 (w μ).2
+  simp only [hwT, diagonalization_symm_apply, map_sum, Submodule.coe_smul_of_tower]
 #align linear_map.is_symmetric.diagonalization_apply_self_apply LinearMap.IsSymmetric.diagonalization_apply_self_apply
 
 end Version1

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

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.

@@ -49,7 +49,7 @@ self-adjoint operator, spectral theorem, diagonalization theorem
 -/
 
 
-variable {𝕜 : Type*} [IsROrC 𝕜]
+variable {𝕜 : Type*} [RCLike 𝕜]
 variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 E _ x y
@@ -211,7 +211,7 @@ for a self-adjoint operator `T` on `E`.
 
 TODO Postcompose with a permutation so that these eigenvalues are listed in increasing order. -/
 noncomputable irreducible_def eigenvalues (i : Fin n) : ℝ :=
-  @IsROrC.re 𝕜 _ <| (hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i
+  @RCLike.re 𝕜 _ <| (hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i
     hT.orthogonalFamily_eigenspaces').val
 #align linear_map.is_symmetric.eigenvalues LinearMap.IsSymmetric.eigenvalues
 
@@ -222,7 +222,7 @@ theorem hasEigenvector_eigenvectorBasis (i : Fin n) :
     (hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i
       hT.orthogonalFamily_eigenspaces').val
   simp_rw [eigenvalues]
-  change HasEigenvector T (IsROrC.re μ) v
+  change HasEigenvector T (RCLike.re μ) v
   have key : HasEigenvector T μ v := by
     have H₁ : v ∈ eigenspace T μ := by
       simp_rw [v, eigenvectorBasis]
@@ -231,8 +231,8 @@ theorem hasEigenvector_eigenvectorBasis (i : Fin n) :
           hT.orthogonalFamily_eigenspaces'
     have H₂ : v ≠ 0 := by simpa using (hT.eigenvectorBasis hn).toBasis.ne_zero i
     exact ⟨H₁, H₂⟩
-  have re_μ : ↑(IsROrC.re μ) = μ := by
-    rw [← IsROrC.conj_eq_iff_re]
+  have re_μ : ↑(RCLike.re μ) = μ := by
+    rw [← RCLike.conj_eq_iff_re]
     exact hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector key)
   simpa [re_μ] using key
 #align linear_map.is_symmetric.has_eigenvector_eigenvector_basis LinearMap.IsSymmetric.hasEigenvector_eigenvectorBasis
@@ -282,25 +282,25 @@ theorem inner_product_apply_eigenvector {μ : 𝕜} {v : E} {T : E →ₗ[𝕜]
 #align inner_product_apply_eigenvector inner_product_apply_eigenvector
 
 theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
-    (hnn : ∀ x : E, 0 ≤ IsROrC.re ⟪x, T x⟫) : 0 ≤ μ := by
+    (hnn : ∀ x : E, 0 ≤ RCLike.re ⟪x, T x⟫) : 0 ≤ μ := by
   obtain ⟨v, hv⟩ := hμ.exists_hasEigenvector
   have hpos : (0 : ℝ) < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2
-  have : IsROrC.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
-    have := congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
+  have : RCLike.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
+    have := congr_arg RCLike.re (inner_product_apply_eigenvector hv.1)
     -- Porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
-    rw [← IsROrC.ofReal_pow, ← IsROrC.ofReal_mul] at this
+    rw [← RCLike.ofReal_pow, ← RCLike.ofReal_mul] at this
     exact mod_cast this
   exact (mul_nonneg_iff_of_pos_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_nonneg_of_nonneg eigenvalue_nonneg_of_nonneg
 
 theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
-    (hnn : ∀ x : E, 0 < IsROrC.re ⟪x, T x⟫) : 0 < μ := by
+    (hnn : ∀ x : E, 0 < RCLike.re ⟪x, T x⟫) : 0 < μ := by
   obtain ⟨v, hv⟩ := hμ.exists_hasEigenvector
   have hpos : (0 : ℝ) < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2
-  have : IsROrC.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
-    have := congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
+  have : RCLike.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
+    have := congr_arg RCLike.re (inner_product_apply_eigenvector hv.1)
     -- Porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
-    rw [← IsROrC.ofReal_pow, ← IsROrC.ofReal_mul] at this
+    rw [← RCLike.ofReal_pow, ← RCLike.ofReal_mul] at this
     exact mod_cast this
   exact (mul_pos_iff_of_pos_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_pos_of_pos eigenvalue_pos_of_pos

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)

@@ -50,7 +50,6 @@ self-adjoint operator, spectral theorem, diagonalization theorem
 
 
 variable {𝕜 : Type*} [IsROrC 𝕜]
-
 variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 E _ x y

This is a very large PR, but it has been reviewed piecemeal already in PRs to the bump/v4.7.0 branch as we update to intermediate nightlies.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: damiano <adomani@gmail.com>

@@ -226,7 +226,7 @@ theorem hasEigenvector_eigenvectorBasis (i : Fin n) :
   change HasEigenvector T (IsROrC.re μ) v
   have key : HasEigenvector T μ v := by
     have H₁ : v ∈ eigenspace T μ := by
-      simp_rw [eigenvectorBasis]
+      simp_rw [v, eigenvectorBasis]
       exact
         hT.direct_sum_isInternal.subordinateOrthonormalBasis_subordinate hn i
           hT.orthogonalFamily_eigenspaces'

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -288,7 +288,7 @@ theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : Has
   have hpos : (0 : ℝ) < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2
   have : IsROrC.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
     have := congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
-    -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
+    -- Porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
     rw [← IsROrC.ofReal_pow, ← IsROrC.ofReal_mul] at this
     exact mod_cast this
   exact (mul_nonneg_iff_of_pos_right hpos).mp (this ▸ hnn v)
@@ -300,7 +300,7 @@ theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenv
   have hpos : (0 : ℝ) < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2
   have : IsROrC.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
     have := congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
-    -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
+    -- Porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
     rw [← IsROrC.ofReal_pow, ← IsROrC.ofReal_mul] at this
     exact mod_cast this
   exact (mul_pos_iff_of_pos_right hpos).mp (this ▸ hnn v)

I had a slightly logic-heavy argument that was nicely simplified by stating this lemma. Also fix a few lemma names.

From LeanAPAP and LeanCamCombi

@@ -291,7 +291,7 @@ theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : Has
     -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
     rw [← IsROrC.ofReal_pow, ← IsROrC.ofReal_mul] at this
     exact mod_cast this
-  exact (zero_le_mul_right hpos).mp (this ▸ hnn v)
+  exact (mul_nonneg_iff_of_pos_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_nonneg_of_nonneg eigenvalue_nonneg_of_nonneg
 
 theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
@@ -303,7 +303,7 @@ theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenv
     -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
     rw [← IsROrC.ofReal_pow, ← IsROrC.ofReal_mul] at this
     exact mod_cast this
-  exact (zero_lt_mul_right hpos).mp (this ▸ hnn v)
+  exact (mul_pos_iff_of_pos_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_pos_of_pos eigenvalue_pos_of_pos
 
 end Nonneg

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

@@ -289,7 +289,7 @@ theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : Has
   have : IsROrC.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
     have := congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
     -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
-    rw [←IsROrC.ofReal_pow, ←IsROrC.ofReal_mul] at this
+    rw [← IsROrC.ofReal_pow, ← IsROrC.ofReal_mul] at this
     exact mod_cast this
   exact (zero_le_mul_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_nonneg_of_nonneg eigenvalue_nonneg_of_nonneg
@@ -301,7 +301,7 @@ theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenv
   have : IsROrC.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
     have := congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
     -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
-    rw [←IsROrC.ofReal_pow, ←IsROrC.ofReal_mul] at this
+    rw [← IsROrC.ofReal_pow, ← IsROrC.ofReal_mul] at this
     exact mod_cast this
   exact (zero_lt_mul_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_pos_of_pos eigenvalue_pos_of_pos

We still have the exact_mod_cast tactic, used in a few places, which somehow (?) works a little bit harder to prevent the expected type influencing the elaboration of the term. I would like to get to the bottom of this, and it will be easier once the only usages of exact_mod_cast are the ones that don't work using the term elaborator by itself.

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

@@ -290,7 +290,7 @@ theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : Has
     have := congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
     -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
     rw [←IsROrC.ofReal_pow, ←IsROrC.ofReal_mul] at this
-    exact_mod_cast this
+    exact mod_cast this
   exact (zero_le_mul_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_nonneg_of_nonneg eigenvalue_nonneg_of_nonneg
 
@@ -302,7 +302,7 @@ theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenv
     have := congr_arg IsROrC.re (inner_product_apply_eigenvector hv.1)
     -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`
     rw [←IsROrC.ofReal_pow, ←IsROrC.ofReal_mul] at this
-    exact_mod_cast this
+    exact mod_cast this
   exact (zero_lt_mul_right hpos).mp (this ▸ hnn v)
 #align eigenvalue_pos_of_pos eigenvalue_pos_of_pos
 

PR contents

This is the supremum of

• #8284
• #8056
• #8023
• #8332
• #8226 (already approved)
• #7834 (already approved)

along with some minor fixes from failures on nightly-testing as Mathlib master is merged into it.

Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.

I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0 branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.

Lean PRs involved in this bump

In particular this includes adjustments for the Lean PRs

• leanprover/lean4#2778
• leanprover/lean4#2790
• leanprover/lean4#2783
• leanprover/lean4#2825
• leanprover/lean4#2722

leanprover/lean4#2778

We can get rid of all the

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)

macros across Mathlib (and in any projects that want to write natural number powers of reals).

leanprover/lean4#2722

Changes the default behaviour of simp to (config := {decide := false}). This makes simp (and consequentially norm_num) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp or norm_num to decide or rfl, or adding (config := {decide := true}).

leanprover/lean4#2783

This changed the behaviour of simp so that simp [f] will only unfold "fully applied" occurrences of f. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true }). We may in future add a syntax for this, e.g. simp [!f]; please provide feedback! In the meantime, we have made the following changes:

• switching to using explicit lemmas that have the intended level of application
• (config := { unfoldPartialApp := true }) in some places, to recover the old behaviour
• Using @[eqns] to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp and Function.flip.

This change in Lean may require further changes down the line (e.g. adding the !f syntax, and/or upstreaming the special treatment for Function.comp and Function.flip, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!

Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>

@@ -276,8 +276,6 @@ end LinearMap
 
 section Nonneg
 
-local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
-
 @[simp]
 theorem inner_product_apply_eigenvector {μ : 𝕜} {v : E} {T : E →ₗ[𝕜] E}
     (h : v ∈ Module.End.eigenspace T μ) : ⟪v, T v⟫ = μ * (‖v‖ : 𝕜) ^ 2 := by

@@ -129,7 +129,7 @@ theorem orthogonalComplement_iSup_eigenspaces_eq_bot : (⨆ μ, eigenspace T μ)
   -- a self-adjoint operator on a nontrivial inner product space has an eigenvalue
   haveI :=
     hT'.subsingleton_of_no_eigenvalue_finiteDimensional hT.orthogonalComplement_iSup_eigenspaces
-  exact Submodule.eq_bot_of_subsingleton _
+  exact Submodule.eq_bot_of_subsingleton
 #align linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot
 
 theorem orthogonalComplement_iSup_eigenspaces_eq_bot' :

We already have noncomputable DecidableEq instances for Real and Complex; putting them here too reduces friction in code generalized to IsROrC K.

@@ -49,7 +49,7 @@ self-adjoint operator, spectral theorem, diagonalization theorem
 -/
 
 
-variable {𝕜 : Type*} [IsROrC 𝕜] [dec_𝕜 : DecidableEq 𝕜]
+variable {𝕜 : Type*} [IsROrC 𝕜]
 
 variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 

Also _root_.map_smul when in the neighbourhood.

@@ -262,8 +262,7 @@ theorem eigenvectorBasis_apply_self_apply (v : E) (i : Fin n) :
       congr_arg (fun v => (hT.eigenvectorBasis hn).repr v i)
         (this ((hT.eigenvectorBasis hn).repr v))
   intro w
-  simp_rw [← OrthonormalBasis.sum_repr_symm, LinearMap.map_sum, LinearMap.map_smul,
-    apply_eigenvectorBasis]
+  simp_rw [← OrthonormalBasis.sum_repr_symm, map_sum, map_smul, apply_eigenvectorBasis]
   apply Fintype.sum_congr
   intro a
   rw [smul_smul, mul_comm]

We clean up heart beat bumps after #6474.

@@ -138,8 +138,6 @@ theorem orthogonalComplement_iSup_eigenspaces_eq_bot' :
     rw [iSup_ne_bot_subtype, hT.orthogonalComplement_iSup_eigenspaces_eq_bot]
 #align linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot' LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot'
 
--- porting note: a modest increast in the `synthInstance.maxHeartbeats`, but we should still fix it.
-set_option synthInstance.maxHeartbeats 23000 in
 /-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` gives
 an internal direct sum decomposition of `E`.
 

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

This has nice performance benefits.

@@ -49,9 +49,9 @@ self-adjoint operator, spectral theorem, diagonalization theorem
 -/
 
 
-variable {𝕜 : Type _} [IsROrC 𝕜] [dec_𝕜 : DecidableEq 𝕜]
+variable {𝕜 : Type*} [IsROrC 𝕜] [dec_𝕜 : DecidableEq 𝕜]
 
-variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 E _ x y
 

@@ -279,8 +279,7 @@ end LinearMap
 
 section Nonneg
 
--- porting note: lean4#2220
-local macro_rules | `($x ^ $y)   => `(HPow.hPow $x $y)
+local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue lean4#2220
 
 @[simp]
 theorem inner_product_apply_eigenvector {μ : 𝕜} {v : E} {T : E →ₗ[𝕜] E}

• depends on: #5966

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

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
-
-! This file was ported from Lean 3 source module analysis.inner_product_space.spectrum
-! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.InnerProductSpace.Rayleigh
 import Mathlib.Analysis.InnerProductSpace.PiL2
 import Mathlib.Algebra.DirectSum.Decomposition
 import Mathlib.LinearAlgebra.Eigenspace.Minpoly
 
+#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
+
 /-! # Spectral theory of self-adjoint operators
 
 This file covers the spectral theory of self-adjoint operators on an inner product space.

@@ -33,7 +33,7 @@ Letting `T` be a self-adjoint operator on a finite-dimensional inner product spa
 * The definition `LinearMap.IsSymmetric.eigenvectorBasis` provides an orthonormal basis for `E`
   consisting of eigenvectors of `T`, with `LinearMap.IsSymmetric.eigenvalues` giving the
   corresponding list of eigenvalues, as real numbers.  The definition
-  `linear_map.is_symmetric.eigenvector_basis` gives the associated linear isometry equivalence
+  `LinearMap.IsSymmetric.eigenvectorBasis` gives the associated linear isometry equivalence
   from `E` to Euclidean space, and the theorem
   `LinearMap.IsSymmetric.eigenvectorBasis_apply_self_apply` states that, when `T` is
   transferred via this equivalence to an operator on Euclidean space, it acts diagonally.

• depends on: #4920

Co-authored-by: Johan Commelin <johan@commelin.net>