linear_algebra.matrix.spectrum ⟷ Mathlib.LinearAlgebra.Matrix.Spectrum

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

@@ -25,7 +25,7 @@ spectral theorem, diagonalization theorem
 
 namespace Matrix
 
-variable {𝕜 : Type _} [IsROrC 𝕜] [DecidableEq 𝕜] {n : Type _} [Fintype n] [DecidableEq n]
+variable {𝕜 : Type _} [RCLike 𝕜] [DecidableEq 𝕜] {n : Type _} [Fintype n] [DecidableEq n]
 
 variable {A : Matrix n n 𝕜}
 
@@ -98,7 +98,7 @@ theorem eigenvectorMatrixInv_apply (i j : n) :
     hA.eigenvectorMatrixInv i j = star (hA.eigenvectorBasis i j) := by
   rw [eigenvector_matrix_inv, Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
     OrthonormalBasis.repr_apply_apply, PiLp.basisFun_apply, WithLp.equiv_symm_single,
-    EuclideanSpace.inner_single_right, one_mul, IsROrC.star_def]
+    EuclideanSpace.inner_single_right, one_mul, RCLike.star_def]
 #align matrix.is_hermitian.eigenvector_matrix_inv_apply Matrix.IsHermitian.eigenvectorMatrixInv_apply
 -/
 
@@ -145,12 +145,12 @@ theorem spectral_theorem :
 #print Matrix.IsHermitian.eigenvalues_eq /-
 theorem eigenvalues_eq (i : n) :
     hA.Eigenvalues i =
-      IsROrC.re (star (hA.eigenvectorMatrixᵀ i) ⬝ᵥ A.mulVec (hA.eigenvectorMatrixᵀ i)) :=
+      RCLike.re (star (hA.eigenvectorMatrixᵀ i) ⬝ᵥ A.mulVec (hA.eigenvectorMatrixᵀ i)) :=
   by
   have := hA.spectral_theorem
   rw [← Matrix.mul_inv_eq_iff_eq_mul_of_invertible] at this
-  have := congr_arg IsROrC.re (congr_fun (congr_fun this i) i)
-  rw [diagonal_apply_eq, IsROrC.ofReal_re, inv_eq_left_inv hA.eigenvector_matrix_mul_inv, ←
+  have := congr_arg RCLike.re (congr_fun (congr_fun this i) i)
+  rw [diagonal_apply_eq, RCLike.ofReal_re, inv_eq_left_inv hA.eigenvector_matrix_mul_inv, ←
     conj_transpose_eigenvector_matrix, mul_mul_apply] at this
   exact this.symm
 #align matrix.is_hermitian.eigenvalues_eq Matrix.IsHermitian.eigenvalues_eq

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

@@ -148,10 +148,10 @@ theorem eigenvalues_eq (i : n) :
       IsROrC.re (star (hA.eigenvectorMatrixᵀ i) ⬝ᵥ A.mulVec (hA.eigenvectorMatrixᵀ i)) :=
   by
   have := hA.spectral_theorem
-  rw [← Matrix.mul_inv_eq_iff_eq_mul_of_invertible] at this 
+  rw [← Matrix.mul_inv_eq_iff_eq_mul_of_invertible] at this
   have := congr_arg IsROrC.re (congr_fun (congr_fun this i) i)
   rw [diagonal_apply_eq, IsROrC.ofReal_re, inv_eq_left_inv hA.eigenvector_matrix_mul_inv, ←
-    conj_transpose_eigenvector_matrix, mul_mul_apply] at this 
+    conj_transpose_eigenvector_matrix, mul_mul_apply] at this
   exact this.symm
 #align matrix.is_hermitian.eigenvalues_eq Matrix.IsHermitian.eigenvalues_eq
 -/

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

@@ -3,8 +3,8 @@ 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.Spectrum
-import Mathbin.LinearAlgebra.Matrix.Hermitian
+import Analysis.InnerProductSpace.Spectrum
+import LinearAlgebra.Matrix.Hermitian
 
 #align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"cff8231f04dfa33fd8f2f45792eebd862ef30cad"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -97,7 +97,7 @@ theorem eigenvectorMatrix_apply (i j : n) : hA.eigenvectorMatrix i j = hA.eigenv
 theorem eigenvectorMatrixInv_apply (i j : n) :
     hA.eigenvectorMatrixInv i j = star (hA.eigenvectorBasis i j) := by
   rw [eigenvector_matrix_inv, Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
-    OrthonormalBasis.repr_apply_apply, PiLp.basisFun_apply, PiLp.equiv_symm_single,
+    OrthonormalBasis.repr_apply_apply, PiLp.basisFun_apply, WithLp.equiv_symm_single,
     EuclideanSpace.inner_single_right, one_mul, IsROrC.star_def]
 #align matrix.is_hermitian.eigenvector_matrix_inv_apply Matrix.IsHermitian.eigenvectorMatrixInv_apply
 -/
@@ -138,7 +138,7 @@ theorem spectral_theorem :
     rfl
   · simp only [diagonal_mul, (· ∘ ·), eigenvalues]
     rw [eigenvector_basis, Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
-      OrthonormalBasis.repr_reindex, eigenvalues₀, PiLp.basisFun_apply, PiLp.equiv_symm_single]
+      OrthonormalBasis.repr_reindex, eigenvalues₀, PiLp.basisFun_apply, WithLp.equiv_symm_single]
 #align matrix.is_hermitian.spectral_theorem Matrix.IsHermitian.spectral_theorem
 -/
 

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

@@ -2,15 +2,12 @@
 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.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.Spectrum
 import Mathbin.LinearAlgebra.Matrix.Hermitian
 
+#align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"cff8231f04dfa33fd8f2f45792eebd862ef30cad"
+
 /-! # Spectral theory of hermitian matrices
 
 > THIS FILE IS SYNCHRONIZED WITH MATHLIB4.

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

@@ -108,7 +108,7 @@ theorem eigenvectorMatrixInv_apply (i j : n) :
 #print Matrix.IsHermitian.conjTranspose_eigenvectorMatrixInv /-
 theorem conjTranspose_eigenvectorMatrixInv : hA.eigenvectorMatrixInvᴴ = hA.eigenvectorMatrix :=
   by
-  ext (i j)
+  ext i j
   rw [conj_transpose_apply, eigenvector_matrix_inv_apply, eigenvector_matrix_apply, star_star]
 #align matrix.is_hermitian.conj_transpose_eigenvector_matrix_inv Matrix.IsHermitian.conjTranspose_eigenvectorMatrixInv
 -/
@@ -128,7 +128,7 @@ theorem spectral_theorem :
     hA.eigenvectorMatrixInv ⬝ A = diagonal (coe ∘ hA.Eigenvalues) ⬝ hA.eigenvectorMatrixInv :=
   by
   rw [eigenvector_matrix_inv, PiLp.basis_toMatrix_basisFun_mul]
-  ext (i j)
+  ext i j
   have := is_hermitian_iff_is_symmetric.1 hA
   convert
     this.diagonalization_basis_apply_self_apply finrank_euclideanSpace (EuclideanSpace.single j 1)

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

@@ -40,36 +40,48 @@ namespace IsHermitian
 
 variable (hA : A.IsHermitian)
 
+#print Matrix.IsHermitian.eigenvalues₀ /-
 /-- The eigenvalues of a hermitian matrix, indexed by `fin (fintype.card n)` where `n` is the index
 type of the matrix. -/
 noncomputable def eigenvalues₀ : Fin (Fintype.card n) → ℝ :=
   (isHermitian_iff_isSymmetric.1 hA).Eigenvalues finrank_euclideanSpace
 #align matrix.is_hermitian.eigenvalues₀ Matrix.IsHermitian.eigenvalues₀
+-/
 
+#print Matrix.IsHermitian.eigenvalues /-
 /-- The eigenvalues of a hermitian matrix, reusing the index `n` of the matrix entries. -/
 noncomputable def eigenvalues : n → ℝ := fun i =>
   hA.eigenvalues₀ <| (Fintype.equivOfCardEq (Fintype.card_fin _)).symm i
 #align matrix.is_hermitian.eigenvalues Matrix.IsHermitian.eigenvalues
+-/
 
+#print Matrix.IsHermitian.eigenvectorBasis /-
 /-- A choice of an orthonormal basis of eigenvectors of a hermitian matrix. -/
 noncomputable def eigenvectorBasis : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n) :=
   ((isHermitian_iff_isSymmetric.1 hA).eigenvectorBasis finrank_euclideanSpace).reindex
     (Fintype.equivOfCardEq (Fintype.card_fin _))
 #align matrix.is_hermitian.eigenvector_basis Matrix.IsHermitian.eigenvectorBasis
+-/
 
+#print Matrix.IsHermitian.eigenvectorMatrix /-
 /-- A matrix whose columns are an orthonormal basis of eigenvectors of a hermitian matrix. -/
 noncomputable def eigenvectorMatrix : Matrix n n 𝕜 :=
   (PiLp.basisFun _ 𝕜 n).toMatrix (eigenvectorBasis hA).toBasis
 #align matrix.is_hermitian.eigenvector_matrix Matrix.IsHermitian.eigenvectorMatrix
+-/
 
+#print Matrix.IsHermitian.eigenvectorMatrixInv /-
 /-- The inverse of `eigenvector_matrix` -/
 noncomputable def eigenvectorMatrixInv : Matrix n n 𝕜 :=
   (eigenvectorBasis hA).toBasis.toMatrix (PiLp.basisFun _ 𝕜 n)
 #align matrix.is_hermitian.eigenvector_matrix_inv Matrix.IsHermitian.eigenvectorMatrixInv
+-/
 
+#print Matrix.IsHermitian.eigenvectorMatrix_mul_inv /-
 theorem eigenvectorMatrix_mul_inv : hA.eigenvectorMatrix ⬝ hA.eigenvectorMatrixInv = 1 := by
   apply Basis.toMatrix_mul_toMatrix_flip
 #align matrix.is_hermitian.eigenvector_matrix_mul_inv Matrix.IsHermitian.eigenvectorMatrix_mul_inv
+-/
 
 noncomputable instance : Invertible hA.eigenvectorMatrixInv :=
   invertibleOfLeftInverse _ _ hA.eigenvectorMatrix_mul_inv
@@ -77,28 +89,37 @@ noncomputable instance : Invertible hA.eigenvectorMatrixInv :=
 noncomputable instance : Invertible hA.eigenvectorMatrix :=
   invertibleOfRightInverse _ _ hA.eigenvectorMatrix_mul_inv
 
+#print Matrix.IsHermitian.eigenvectorMatrix_apply /-
 theorem eigenvectorMatrix_apply (i j : n) : hA.eigenvectorMatrix i j = hA.eigenvectorBasis j i := by
   simp_rw [eigenvector_matrix, Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis,
     PiLp.basisFun_repr]
 #align matrix.is_hermitian.eigenvector_matrix_apply Matrix.IsHermitian.eigenvectorMatrix_apply
+-/
 
+#print Matrix.IsHermitian.eigenvectorMatrixInv_apply /-
 theorem eigenvectorMatrixInv_apply (i j : n) :
     hA.eigenvectorMatrixInv i j = star (hA.eigenvectorBasis i j) := by
   rw [eigenvector_matrix_inv, Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
     OrthonormalBasis.repr_apply_apply, PiLp.basisFun_apply, PiLp.equiv_symm_single,
     EuclideanSpace.inner_single_right, one_mul, IsROrC.star_def]
 #align matrix.is_hermitian.eigenvector_matrix_inv_apply Matrix.IsHermitian.eigenvectorMatrixInv_apply
+-/
 
+#print Matrix.IsHermitian.conjTranspose_eigenvectorMatrixInv /-
 theorem conjTranspose_eigenvectorMatrixInv : hA.eigenvectorMatrixInvᴴ = hA.eigenvectorMatrix :=
   by
   ext (i j)
   rw [conj_transpose_apply, eigenvector_matrix_inv_apply, eigenvector_matrix_apply, star_star]
 #align matrix.is_hermitian.conj_transpose_eigenvector_matrix_inv Matrix.IsHermitian.conjTranspose_eigenvectorMatrixInv
+-/
 
+#print Matrix.IsHermitian.conjTranspose_eigenvectorMatrix /-
 theorem conjTranspose_eigenvectorMatrix : hA.eigenvectorMatrixᴴ = hA.eigenvectorMatrixInv := by
   rw [← conj_transpose_eigenvector_matrix_inv, conj_transpose_conj_transpose]
 #align matrix.is_hermitian.conj_transpose_eigenvector_matrix Matrix.IsHermitian.conjTranspose_eigenvectorMatrix
+-/
 
+#print Matrix.IsHermitian.spectral_theorem /-
 /-- *Diagonalization theorem*, *spectral theorem* for matrices; A hermitian matrix can be
 diagonalized by a change of basis.
 
@@ -122,7 +143,9 @@ theorem spectral_theorem :
     rw [eigenvector_basis, Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
       OrthonormalBasis.repr_reindex, eigenvalues₀, PiLp.basisFun_apply, PiLp.equiv_symm_single]
 #align matrix.is_hermitian.spectral_theorem Matrix.IsHermitian.spectral_theorem
+-/
 
+#print Matrix.IsHermitian.eigenvalues_eq /-
 theorem eigenvalues_eq (i : n) :
     hA.Eigenvalues i =
       IsROrC.re (star (hA.eigenvectorMatrixᵀ i) ⬝ᵥ A.mulVec (hA.eigenvectorMatrixᵀ i)) :=
@@ -134,13 +157,16 @@ theorem eigenvalues_eq (i : n) :
     conj_transpose_eigenvector_matrix, mul_mul_apply] at this 
   exact this.symm
 #align matrix.is_hermitian.eigenvalues_eq Matrix.IsHermitian.eigenvalues_eq
+-/
 
+#print Matrix.IsHermitian.det_eq_prod_eigenvalues /-
 /-- The determinant of a hermitian matrix is the product of its eigenvalues. -/
 theorem det_eq_prod_eigenvalues : det A = ∏ i, hA.Eigenvalues i :=
   by
   apply mul_left_cancel₀ (det_ne_zero_of_left_inverse (eigenvector_matrix_mul_inv hA))
   rw [← det_mul, spectral_theorem, det_mul, mul_comm, det_diagonal]
 #align matrix.is_hermitian.det_eq_prod_eigenvalues Matrix.IsHermitian.det_eq_prod_eigenvalues
+-/
 
 end IsHermitian
 

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: Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module linear_algebra.matrix.spectrum
-! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
+! leanprover-community/mathlib commit cff8231f04dfa33fd8f2f45792eebd862ef30cad
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.LinearAlgebra.Matrix.Hermitian
 
 /-! # Spectral theory of hermitian matrices
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file proves the spectral theorem for matrices. The proof of the spectral theorem is based on
 the spectral theorem for linear maps (`diagonalization_basis_apply_self_apply`).
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -106,8 +106,9 @@ theorem spectral_theorem :
   rw [eigenvector_matrix_inv, PiLp.basis_toMatrix_basisFun_mul]
   ext (i j)
   have := is_hermitian_iff_is_symmetric.1 hA
-  convert this.diagonalization_basis_apply_self_apply finrank_euclideanSpace
-      (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq (Fintype.card_fin _)).symm i) using
+  convert
+    this.diagonalization_basis_apply_self_apply finrank_euclideanSpace (EuclideanSpace.single j 1)
+      ((Fintype.equivOfCardEq (Fintype.card_fin _)).symm i) using
     1
   · dsimp only [EuclideanSpace.single, to_euclidean_lin_pi_Lp_equiv_symm, to_lin'_apply,
       Matrix.of_apply, is_hermitian.eigenvector_basis]

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

@@ -124,10 +124,10 @@ theorem eigenvalues_eq (i : n) :
       IsROrC.re (star (hA.eigenvectorMatrixᵀ i) ⬝ᵥ A.mulVec (hA.eigenvectorMatrixᵀ i)) :=
   by
   have := hA.spectral_theorem
-  rw [← Matrix.mul_inv_eq_iff_eq_mul_of_invertible] at this
+  rw [← Matrix.mul_inv_eq_iff_eq_mul_of_invertible] at this 
   have := congr_arg IsROrC.re (congr_fun (congr_fun this i) i)
   rw [diagonal_apply_eq, IsROrC.ofReal_re, inv_eq_left_inv hA.eigenvector_matrix_mul_inv, ←
-    conj_transpose_eigenvector_matrix, mul_mul_apply] at this
+    conj_transpose_eigenvector_matrix, mul_mul_apply] at this 
   exact this.symm
 #align matrix.is_hermitian.eigenvalues_eq Matrix.IsHermitian.eigenvalues_eq
 

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

@@ -29,9 +29,9 @@ variable {𝕜 : Type _} [IsROrC 𝕜] [DecidableEq 𝕜] {n : Type _} [Fintype
 
 variable {A : Matrix n n 𝕜}
 
-open Matrix
+open scoped Matrix
 
-open BigOperators
+open scoped BigOperators
 
 namespace IsHermitian
 

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

@@ -126,7 +126,7 @@ theorem eigenvalues_eq (i : n) :
   have := hA.spectral_theorem
   rw [← Matrix.mul_inv_eq_iff_eq_mul_of_invertible] at this
   have := congr_arg IsROrC.re (congr_fun (congr_fun this i) i)
-  rw [diagonal_apply_eq, IsROrC.of_real_re, inv_eq_left_inv hA.eigenvector_matrix_mul_inv, ←
+  rw [diagonal_apply_eq, IsROrC.ofReal_re, inv_eq_left_inv hA.eigenvector_matrix_mul_inv, ←
     conj_transpose_eigenvector_matrix, mul_mul_apply] at this
   exact this.symm
 #align matrix.is_hermitian.eigenvalues_eq Matrix.IsHermitian.eigenvalues_eq

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: Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module linear_algebra.matrix.spectrum
-! leanprover-community/mathlib commit 9f0d61b4475e3c3cba6636ab51cdb1f3949d2e1d
+! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -114,8 +114,8 @@ theorem spectral_theorem :
     simp_rw [mul_vec_single, mul_one, OrthonormalBasis.coe_toBasis_repr_apply,
       OrthonormalBasis.repr_reindex]
     rfl
-  · simp only [diagonal_mul, (· ∘ ·), eigenvalues, eigenvector_basis]
-    rw [Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
+  · simp only [diagonal_mul, (· ∘ ·), eigenvalues]
+    rw [eigenvector_basis, Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
       OrthonormalBasis.repr_reindex, eigenvalues₀, PiLp.basisFun_apply, PiLp.equiv_symm_single]
 #align matrix.is_hermitian.spectral_theorem Matrix.IsHermitian.spectral_theorem
 

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.spectrum
-! leanprover-community/mathlib commit 2f4cdce0c2f2f3b8cd58f05d556d03b468e1eb2e
+! leanprover-community/mathlib commit 9f0d61b4475e3c3cba6636ab51cdb1f3949d2e1d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -105,15 +105,14 @@ theorem spectral_theorem :
   by
   rw [eigenvector_matrix_inv, PiLp.basis_toMatrix_basisFun_mul]
   ext (i j)
-  have : LinearMap.IsSymmetric _ := is_hermitian_iff_is_symmetric.1 hA
+  have := is_hermitian_iff_is_symmetric.1 hA
   convert this.diagonalization_basis_apply_self_apply finrank_euclideanSpace
-      (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq (Fintype.card_fin _)).symm i)
-  · dsimp only [LinearEquiv.conj_apply_apply, PiLp.linearEquiv_apply, PiLp.linearEquiv_symm_apply,
-      PiLp.equiv_single, LinearMap.stdBasis, LinearMap.coe_single, PiLp.equiv_symm_single,
-      LinearEquiv.symm_symm, eigenvector_basis, to_lin'_apply]
-    simp only [Basis.toMatrix, Basis.coe_toOrthonormalBasis_repr, Basis.equivFun_apply]
-    simp_rw [OrthonormalBasis.coe_toBasis_repr_apply, OrthonormalBasis.repr_reindex,
-      LinearEquiv.symm_symm, PiLp.linearEquiv_apply, PiLp.equiv_single, mul_vec_single, mul_one]
+      (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq (Fintype.card_fin _)).symm i) using
+    1
+  · dsimp only [EuclideanSpace.single, to_euclidean_lin_pi_Lp_equiv_symm, to_lin'_apply,
+      Matrix.of_apply, is_hermitian.eigenvector_basis]
+    simp_rw [mul_vec_single, mul_one, OrthonormalBasis.coe_toBasis_repr_apply,
+      OrthonormalBasis.repr_reindex]
     rfl
   · simp only [diagonal_mul, (· ∘ ·), eigenvalues, eigenvector_basis]
     rw [Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,

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

@@ -106,9 +106,8 @@ theorem spectral_theorem :
   rw [eigenvector_matrix_inv, PiLp.basis_toMatrix_basisFun_mul]
   ext (i j)
   have : LinearMap.IsSymmetric _ := is_hermitian_iff_is_symmetric.1 hA
-  convert
-    this.diagonalization_basis_apply_self_apply finrank_euclideanSpace (EuclideanSpace.single j 1)
-      ((Fintype.equivOfCardEq (Fintype.card_fin _)).symm i)
+  convert this.diagonalization_basis_apply_self_apply finrank_euclideanSpace
+      (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq (Fintype.card_fin _)).symm i)
   · dsimp only [LinearEquiv.conj_apply_apply, PiLp.linearEquiv_apply, PiLp.linearEquiv_symm_apply,
       PiLp.equiv_single, LinearMap.stdBasis, LinearMap.coe_single, PiLp.equiv_symm_single,
       LinearEquiv.symm_symm, eigenvector_basis, to_lin'_apply]

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.

@@ -24,7 +24,7 @@ spectral theorem, diagonalization theorem
 
 namespace Matrix
 
-variable {𝕜 : Type*} [IsROrC 𝕜] {n : Type*} [Fintype n]
+variable {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
 variable {A : Matrix n n 𝕜}
 
 open scoped BigOperators
@@ -87,7 +87,7 @@ theorem eigenvectorMatrixInv_apply (i j : n) :
     hA.eigenvectorMatrixInv i j = star (hA.eigenvectorBasis i j) := by
   rw [eigenvectorMatrixInv, Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
     OrthonormalBasis.repr_apply_apply, PiLp.basisFun_apply, WithLp.equiv_symm_single,
-    EuclideanSpace.inner_single_right, one_mul, IsROrC.star_def]
+    EuclideanSpace.inner_single_right, one_mul, RCLike.star_def]
 #align matrix.is_hermitian.eigenvector_matrix_inv_apply Matrix.IsHermitian.eigenvectorMatrixInv_apply
 
 theorem conjTranspose_eigenvectorMatrixInv : hA.eigenvectorMatrixInvᴴ = hA.eigenvectorMatrix := by
@@ -123,11 +123,11 @@ theorem spectral_theorem :
 
 theorem eigenvalues_eq (i : n) :
     hA.eigenvalues i =
-      IsROrC.re (star (hA.eigenvectorMatrixᵀ i) ⬝ᵥ A *ᵥ hA.eigenvectorMatrixᵀ i) := by
+      RCLike.re (star (hA.eigenvectorMatrixᵀ i) ⬝ᵥ A *ᵥ hA.eigenvectorMatrixᵀ i) := by
   have := hA.spectral_theorem
   rw [← @Matrix.mul_inv_eq_iff_eq_mul_of_invertible (A := hA.eigenvectorMatrixInv)] at this
-  have := congr_arg IsROrC.re (congr_fun (congr_fun this i) i)
-  rw [diagonal_apply_eq, Function.comp_apply, IsROrC.ofReal_re,
+  have := congr_arg RCLike.re (congr_fun (congr_fun this i) i)
+  rw [diagonal_apply_eq, Function.comp_apply, RCLike.ofReal_re,
     inv_eq_left_inv hA.eigenvectorMatrix_mul_inv, ← conjTranspose_eigenvectorMatrix, mul_mul_apply]
     at this
   exact this.symm
@@ -170,7 +170,7 @@ lemma mulVec_eigenvectorBasis (i : n) :
   have := congr_fun (congr_fun this j) i
   simp only [mul_diagonal, Function.comp_apply] at this
   convert this using 1
-  rw [mul_comm, Pi.smul_apply, IsROrC.real_smul_eq_coe_mul, hA.eigenvectorMatrix_apply]
+  rw [mul_comm, Pi.smul_apply, RCLike.real_smul_eq_coe_mul, hA.eigenvectorMatrix_apply]
 
 end DecidableEq
 
@@ -181,7 +181,7 @@ lemma exists_eigenvector_of_ne_zero (hA : IsHermitian A) (h_ne : A ≠ 0) :
   have : hA.eigenvalues ≠ 0 := by
     contrapose! h_ne
     have := hA.spectral_theorem'
-    rwa [h_ne, Pi.comp_zero, IsROrC.ofReal_zero, (by rfl : Function.const n (0 : 𝕜) = fun _ ↦ 0),
+    rwa [h_ne, Pi.comp_zero, RCLike.ofReal_zero, (by rfl : Function.const n (0 : 𝕜) = fun _ ↦ 0),
       diagonal_zero, mul_zero, zero_mul] at this
   obtain ⟨i, hi⟩ := Function.ne_iff.mp this
   exact ⟨_, _, hi, hA.eigenvectorBasis.orthonormal.ne_zero i, hA.mulVec_eigenvectorBasis i⟩

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)

@@ -25,7 +25,6 @@ spectral theorem, diagonalization theorem
 namespace Matrix
 
 variable {𝕜 : Type*} [IsROrC 𝕜] {n : Type*} [Fintype n]
-
 variable {A : Matrix n n 𝕜}
 
 open scoped BigOperators
@@ -35,7 +34,6 @@ namespace IsHermitian
 section DecidableEq
 
 variable [DecidableEq n]
-
 variable (hA : A.IsHermitian)
 
 /-- The eigenvalues of a hermitian matrix, indexed by `Fin (Fintype.card n)` where `n` is the index

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -180,8 +180,8 @@ end DecidableEq
 lemma exists_eigenvector_of_ne_zero (hA : IsHermitian A) (h_ne : A ≠ 0) :
     ∃ (v : n → 𝕜) (t : ℝ), t ≠ 0 ∧ v ≠ 0 ∧ A *ᵥ v = t • v := by
   classical
-  have : hA.eigenvalues ≠ 0
-  · contrapose! h_ne
+  have : hA.eigenvalues ≠ 0 := by
+    contrapose! h_ne
     have := hA.spectral_theorem'
     rwa [h_ne, Pi.comp_zero, IsROrC.ofReal_zero, (by rfl : Function.const n (0 : 𝕜) = fun _ ↦ 0),
       diagonal_zero, mul_zero, zero_mul] at this

Zulip discussion: https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Notation.20for.20mul_vec.20and.20vec_mul

Co-authored-by: Martin Dvorak <mdvorak@ista.ac.at>

@@ -125,7 +125,7 @@ theorem spectral_theorem :
 
 theorem eigenvalues_eq (i : n) :
     hA.eigenvalues i =
-      IsROrC.re (star (hA.eigenvectorMatrixᵀ i) ⬝ᵥ A.mulVec (hA.eigenvectorMatrixᵀ i)) := by
+      IsROrC.re (star (hA.eigenvectorMatrixᵀ i) ⬝ᵥ A *ᵥ hA.eigenvectorMatrixᵀ i) := by
   have := hA.spectral_theorem
   rw [← @Matrix.mul_inv_eq_iff_eq_mul_of_invertible (A := hA.eigenvectorMatrixInv)] at this
   have := congr_arg IsROrC.re (congr_fun (congr_fun this i) i)
@@ -165,7 +165,7 @@ lemma rank_eq_card_non_zero_eigs : A.rank = Fintype.card {i // hA.eigenvalues i
 
 /-- The entries of `eigenvectorBasis` are eigenvectors. -/
 lemma mulVec_eigenvectorBasis (i : n) :
-    mulVec A (hA.eigenvectorBasis i) = hA.eigenvalues i • hA.eigenvectorBasis i := by
+    A *ᵥ hA.eigenvectorBasis i = hA.eigenvalues i • hA.eigenvectorBasis i := by
   have := congr_arg (· * hA.eigenvectorMatrix) hA.spectral_theorem'
   simp only [mul_assoc, mul_eq_one_comm.mp hA.eigenvectorMatrix_mul_inv, mul_one] at this
   ext1 j
@@ -178,7 +178,7 @@ end DecidableEq
 
 /-- A nonzero Hermitian matrix has an eigenvector with nonzero eigenvalue. -/
 lemma exists_eigenvector_of_ne_zero (hA : IsHermitian A) (h_ne : A ≠ 0) :
-    ∃ (v : n → 𝕜) (t : ℝ), t ≠ 0 ∧ v ≠ 0 ∧ mulVec A v = t • v := by
+    ∃ (v : n → 𝕜) (t : ℝ), t ≠ 0 ∧ v ≠ 0 ∧ A *ᵥ v = t • v := by
   classical
   have : hA.eigenvalues ≠ 0
   · contrapose! h_ne

Found by a linter from #10235.

@@ -24,16 +24,18 @@ spectral theorem, diagonalization theorem
 
 namespace Matrix
 
-variable {𝕜 : Type*} [IsROrC 𝕜] {n : Type*} [Fintype n] [DecidableEq n]
+variable {𝕜 : Type*} [IsROrC 𝕜] {n : Type*} [Fintype n]
 
 variable {A : Matrix n n 𝕜}
 
-open scoped Matrix
-
 open scoped BigOperators
 
 namespace IsHermitian
 
+section DecidableEq
+
+variable [DecidableEq n]
+
 variable (hA : A.IsHermitian)
 
 /-- The eigenvalues of a hermitian matrix, indexed by `Fin (Fintype.card n)` where `n` is the index
@@ -172,9 +174,12 @@ lemma mulVec_eigenvectorBasis (i : n) :
   convert this using 1
   rw [mul_comm, Pi.smul_apply, IsROrC.real_smul_eq_coe_mul, hA.eigenvectorMatrix_apply]
 
+end DecidableEq
+
 /-- A nonzero Hermitian matrix has an eigenvector with nonzero eigenvalue. -/
-lemma exists_eigenvector_of_ne_zero (h_ne : A ≠ 0) :
+lemma exists_eigenvector_of_ne_zero (hA : IsHermitian A) (h_ne : A ≠ 0) :
     ∃ (v : n → 𝕜) (t : ℝ), t ≠ 0 ∧ v ≠ 0 ∧ mulVec A v = t • v := by
+  classical
   have : hA.eigenvalues ≠ 0
   · contrapose! h_ne
     have := hA.spectral_theorem'

See Zulip thread for the discussion.

@@ -142,7 +142,7 @@ theorem det_eq_prod_eigenvalues : det A = ∏ i, (hA.eigenvalues i : 𝕜) := by
 
 /-- *spectral theorem* (Alternate form for convenience) A hermitian matrix can be can be
 replaced by a diagonal matrix sandwiched between the eigenvector matrices. This alternate form
-allows direct rewriting of A since: $ A = V D V⁻¹$ -/
+allows direct rewriting of A since: <| A = V D V⁻¹$ -/
 lemma spectral_theorem' :
     A = hA.eigenvectorMatrix * diagonal ((↑) ∘ hA.eigenvalues) * hA.eigenvectorMatrixInv := by
   simpa [ ← Matrix.mul_assoc, hA.eigenvectorMatrix_mul_inv, Matrix.one_mul] using

Subset of #9319

@@ -81,7 +81,7 @@ theorem eigenvectorMatrix_apply (i j : n) : hA.eigenvectorMatrix i j = hA.eigenv
 /-- The columns of `Matrix.IsHermitian.eigenVectorMatrix` form the basis-/
 theorem transpose_eigenvectorMatrix_apply (i : n) :
     hA.eigenvectorMatrixᵀ i = hA.eigenvectorBasis i :=
-  funext <| fun j => eigenvectorMatrix_apply hA j i
+  funext fun j => eigenvectorMatrix_apply hA j i
 
 theorem eigenvectorMatrixInv_apply (i j : n) :
     hA.eigenvectorMatrixInv i j = star (hA.eigenvectorBasis i j) := by

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>

@@ -161,6 +161,28 @@ lemma rank_eq_rank_diagonal : A.rank = (Matrix.diagonal hA.eigenvalues).rank :=
 lemma rank_eq_card_non_zero_eigs : A.rank = Fintype.card {i // hA.eigenvalues i ≠ 0} := by
   rw [rank_eq_rank_diagonal hA, Matrix.rank_diagonal]
 
+/-- The entries of `eigenvectorBasis` are eigenvectors. -/
+lemma mulVec_eigenvectorBasis (i : n) :
+    mulVec A (hA.eigenvectorBasis i) = hA.eigenvalues i • hA.eigenvectorBasis i := by
+  have := congr_arg (· * hA.eigenvectorMatrix) hA.spectral_theorem'
+  simp only [mul_assoc, mul_eq_one_comm.mp hA.eigenvectorMatrix_mul_inv, mul_one] at this
+  ext1 j
+  have := congr_fun (congr_fun this j) i
+  simp only [mul_diagonal, Function.comp_apply] at this
+  convert this using 1
+  rw [mul_comm, Pi.smul_apply, IsROrC.real_smul_eq_coe_mul, hA.eigenvectorMatrix_apply]
+
+/-- A nonzero Hermitian matrix has an eigenvector with nonzero eigenvalue. -/
+lemma exists_eigenvector_of_ne_zero (h_ne : A ≠ 0) :
+    ∃ (v : n → 𝕜) (t : ℝ), t ≠ 0 ∧ v ≠ 0 ∧ mulVec A v = t • v := by
+  have : hA.eigenvalues ≠ 0
+  · contrapose! h_ne
+    have := hA.spectral_theorem'
+    rwa [h_ne, Pi.comp_zero, IsROrC.ofReal_zero, (by rfl : Function.const n (0 : 𝕜) = fun _ ↦ 0),
+      diagonal_zero, mul_zero, zero_mul] at this
+  obtain ⟨i, hi⟩ := Function.ne_iff.mp this
+  exact ⟨_, _, hi, hA.eigenvectorBasis.orthonormal.ne_zero i, hA.mulVec_eigenvectorBasis i⟩
+
 end IsHermitian
 
 end Matrix

@@ -99,7 +99,7 @@ theorem conjTranspose_eigenvectorMatrix : hA.eigenvectorMatrixᴴ = hA.eigenvect
   rw [← conjTranspose_eigenvectorMatrixInv, conjTranspose_conjTranspose]
 #align matrix.is_hermitian.conj_transpose_eigenvector_matrix Matrix.IsHermitian.conjTranspose_eigenvectorMatrix
 
-/-- *Diagonalization theorem*, *spectral theorem* for matrices; A hermitian matrix can be
+/-- **Diagonalization theorem**, **spectral theorem** for matrices; A hermitian matrix can be
 diagonalized by a change of basis.
 
 For the spectral theorem on linear maps, see

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

@@ -24,7 +24,7 @@ spectral theorem, diagonalization theorem
 
 namespace Matrix
 
-variable {𝕜 : Type*} [IsROrC 𝕜] [DecidableEq 𝕜] {n : Type*} [Fintype n] [DecidableEq n]
+variable {𝕜 : Type*} [IsROrC 𝕜] {n : Type*} [Fintype n] [DecidableEq n]
 
 variable {A : Matrix n n 𝕜}
 

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.

@@ -86,7 +86,7 @@ theorem transpose_eigenvectorMatrix_apply (i : n) :
 theorem eigenvectorMatrixInv_apply (i j : n) :
     hA.eigenvectorMatrixInv i j = star (hA.eigenvectorBasis i j) := by
   rw [eigenvectorMatrixInv, Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
-    OrthonormalBasis.repr_apply_apply, PiLp.basisFun_apply, PiLp.equiv_symm_single,
+    OrthonormalBasis.repr_apply_apply, PiLp.basisFun_apply, WithLp.equiv_symm_single,
     EuclideanSpace.inner_single_right, one_mul, IsROrC.star_def]
 #align matrix.is_hermitian.eigenvector_matrix_inv_apply Matrix.IsHermitian.eigenvectorMatrixInv_apply
 
@@ -118,7 +118,7 @@ theorem spectral_theorem :
     rfl
   · simp only [diagonal_mul, (· ∘ ·), eigenvalues]
     rw [eigenvectorBasis, Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
-      OrthonormalBasis.repr_reindex, eigenvalues₀, PiLp.basisFun_apply, PiLp.equiv_symm_single]
+      OrthonormalBasis.repr_reindex, eigenvalues₀, PiLp.basisFun_apply, WithLp.equiv_symm_single]
 #align matrix.is_hermitian.spectral_theorem Matrix.IsHermitian.spectral_theorem
 
 theorem eigenvalues_eq (i : n) :

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).

@@ -63,7 +63,7 @@ noncomputable def eigenvectorMatrixInv : Matrix n n 𝕜 :=
   (eigenvectorBasis hA).toBasis.toMatrix (PiLp.basisFun _ 𝕜 n)
 #align matrix.is_hermitian.eigenvector_matrix_inv Matrix.IsHermitian.eigenvectorMatrixInv
 
-theorem eigenvectorMatrix_mul_inv : hA.eigenvectorMatrix ⬝ hA.eigenvectorMatrixInv = 1 := by
+theorem eigenvectorMatrix_mul_inv : hA.eigenvectorMatrix * hA.eigenvectorMatrixInv = 1 := by
   apply Basis.toMatrix_mul_toMatrix_flip
 #align matrix.is_hermitian.eigenvector_matrix_mul_inv Matrix.IsHermitian.eigenvectorMatrix_mul_inv
 
@@ -105,7 +105,7 @@ diagonalized by a change of basis.
 For the spectral theorem on linear maps, see
 `LinearMap.IsSymmetric.eigenvectorBasis_apply_self_apply`. -/
 theorem spectral_theorem :
-    hA.eigenvectorMatrixInv ⬝ A = diagonal ((↑) ∘ hA.eigenvalues) ⬝ hA.eigenvectorMatrixInv := by
+    hA.eigenvectorMatrixInv * A = diagonal ((↑) ∘ hA.eigenvalues) * hA.eigenvectorMatrixInv := by
   rw [eigenvectorMatrixInv, PiLp.basis_toMatrix_basisFun_mul]
   ext i j
   have := isHermitian_iff_isSymmetric.1 hA
@@ -144,9 +144,9 @@ theorem det_eq_prod_eigenvalues : det A = ∏ i, (hA.eigenvalues i : 𝕜) := by
 replaced by a diagonal matrix sandwiched between the eigenvector matrices. This alternate form
 allows direct rewriting of A since: $ A = V D V⁻¹$ -/
 lemma spectral_theorem' :
-    A = hA.eigenvectorMatrix ⬝ diagonal ((↑) ∘ hA.eigenvalues) ⬝ hA.eigenvectorMatrixInv := by
+    A = hA.eigenvectorMatrix * diagonal ((↑) ∘ hA.eigenvalues) * hA.eigenvectorMatrixInv := by
   simpa [ ← Matrix.mul_assoc, hA.eigenvectorMatrix_mul_inv, Matrix.one_mul] using
-    congr_arg (hA.eigenvectorMatrix ⬝ ·) hA.spectral_theorem
+    congr_arg (hA.eigenvectorMatrix * ·) hA.spectral_theorem
 
 /-- rank of a hermitian matrix is the rank of after diagonalization by the eigenvector matrix -/
 lemma rank_eq_rank_diagonal : A.rank = (Matrix.diagonal hA.eigenvalues).rank := by

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

This has nice performance benefits.

@@ -24,7 +24,7 @@ spectral theorem, diagonalization theorem
 
 namespace Matrix
 
-variable {𝕜 : Type _} [IsROrC 𝕜] [DecidableEq 𝕜] {n : Type _} [Fintype n] [DecidableEq n]
+variable {𝕜 : Type*} [IsROrC 𝕜] [DecidableEq 𝕜] {n : Type*} [Fintype n] [DecidableEq n]
 
 variable {A : Matrix n n 𝕜}
 

This PR provides the lemmas:

• IsHermitian.spectral_theorem': a slightly modified spectral_theorem such that the hermitian matrix can be directly replaced. $$A = VDV^{-1}\quad D = \text{diag}(d)$$
• IsHermitian.rank_eq_count_non_zero_eigs: the rank of a hermitian matrix is the number of non-zero eigenvalues of that matrix $$\text{rank}(A) = \text{card} \lbrace \quad i \quad | \quad d_i \neq 0 \rbrace$$
• IsHermitian.rank_eq_rank_diagonal: the rank of a hermitian matrix is the same as its rank after diagonalization by the eigenvector matrix $$\text{rank}(A) = \text{rank}(D)$$
• Matrix.rank_diagonal: the rank of a diagonal matrix is the number of non-zero diagonal elements. Note that this was previously the name of a lemma about LinearMaps we take that name (but keep the align to avoid disrupting the tooling). $$\text{rank}(V) = \text{Diag}(v) \implies \text{rank}(V) = \text{card} \lbrace \quad i \quad | \quad v_i \neq 0 \rbrace$$
• LinearMap.rank_diagonal: the rank of a linear map that is formed from a diagonal matrix is the number of non-zero diagonal entries. This was previously called Matrix.rank_diagonal.

Co-authored-by: Mohanad Ahmed <m.a.m.elhassan@gmail.com>

@@ -4,6 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
 -/
 import Mathlib.Analysis.InnerProductSpace.Spectrum
+import Mathlib.Data.Matrix.Rank
+import Mathlib.LinearAlgebra.Matrix.Diagonal
 import Mathlib.LinearAlgebra.Matrix.Hermitian
 
 #align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
@@ -138,6 +140,27 @@ theorem det_eq_prod_eigenvalues : det A = ∏ i, (hA.eigenvalues i : 𝕜) := by
   simp_rw [Function.comp_apply]
 #align matrix.is_hermitian.det_eq_prod_eigenvalues Matrix.IsHermitian.det_eq_prod_eigenvalues
 
+/-- *spectral theorem* (Alternate form for convenience) A hermitian matrix can be can be
+replaced by a diagonal matrix sandwiched between the eigenvector matrices. This alternate form
+allows direct rewriting of A since: $ A = V D V⁻¹$ -/
+lemma spectral_theorem' :
+    A = hA.eigenvectorMatrix ⬝ diagonal ((↑) ∘ hA.eigenvalues) ⬝ hA.eigenvectorMatrixInv := by
+  simpa [ ← Matrix.mul_assoc, hA.eigenvectorMatrix_mul_inv, Matrix.one_mul] using
+    congr_arg (hA.eigenvectorMatrix ⬝ ·) hA.spectral_theorem
+
+/-- rank of a hermitian matrix is the rank of after diagonalization by the eigenvector matrix -/
+lemma rank_eq_rank_diagonal : A.rank = (Matrix.diagonal hA.eigenvalues).rank := by
+  conv_lhs => rw [hA.spectral_theorem']
+  have hE := isUnit_det_of_invertible (hA.eigenvectorMatrix)
+  have hiE := isUnit_det_of_invertible (hA.eigenvectorMatrixInv)
+  simp only [rank_mul_eq_right_of_isUnit_det hA.eigenvectorMatrix _ hE,
+    rank_mul_eq_left_of_isUnit_det hA.eigenvectorMatrixInv _ hiE,
+    rank_diagonal, Function.comp_apply, ne_eq, algebraMap.lift_map_eq_zero_iff]
+
+/-- rank of a hermitian matrix is the number of nonzero eigenvalues of the hermitian matrix -/
+lemma rank_eq_card_non_zero_eigs : A.rank = Fintype.card {i // hA.eigenvalues i ≠ 0} := by
+  rw [rank_eq_rank_diagonal hA, Matrix.rank_diagonal]
+
 end IsHermitian
 
 end Matrix

Two lemmas:

• Matrix.PosDef.eigenvalues_pos: $$A \in k^{n\times n}, A > 0 \implies \lambda (A) >0$$
• Matrix.PosSemidef.eigenvalues_nonneg: $$A \in k^{n\times n}, A \geq 0 \implies \lambda (A) \geq 0$$

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

@@ -76,6 +76,11 @@ theorem eigenvectorMatrix_apply (i j : n) : hA.eigenvectorMatrix i j = hA.eigenv
     PiLp.basisFun_repr]
 #align matrix.is_hermitian.eigenvector_matrix_apply Matrix.IsHermitian.eigenvectorMatrix_apply
 
+/-- The columns of `Matrix.IsHermitian.eigenVectorMatrix` form the basis-/
+theorem transpose_eigenvectorMatrix_apply (i : n) :
+    hA.eigenvectorMatrixᵀ i = hA.eigenvectorBasis i :=
+  funext <| fun j => eigenvectorMatrix_apply hA j i
+
 theorem eigenvectorMatrixInv_apply (i j : n) :
     hA.eigenvectorMatrixInv i j = star (hA.eigenvectorBasis i j) := by
   rw [eigenvectorMatrixInv, Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 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.spectrum
-! leanprover-community/mathlib commit 46b633fd842bef9469441c0209906f6dddd2b4f5
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.InnerProductSpace.Spectrum
 import Mathlib.LinearAlgebra.Matrix.Hermitian
 
+#align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
+
 /-! # Spectral theory of hermitian matrices
 
 This file proves the spectral theorem for matrices. The proof of the spectral theorem is based on

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>

@@ -87,7 +87,7 @@ theorem eigenvectorMatrixInv_apply (i j : n) :
 #align matrix.is_hermitian.eigenvector_matrix_inv_apply Matrix.IsHermitian.eigenvectorMatrixInv_apply
 
 theorem conjTranspose_eigenvectorMatrixInv : hA.eigenvectorMatrixInvᴴ = hA.eigenvectorMatrix := by
-  ext (i j)
+  ext i j
   rw [conjTranspose_apply, eigenvectorMatrixInv_apply, eigenvectorMatrix_apply, star_star]
 #align matrix.is_hermitian.conj_transpose_eigenvector_matrix_inv Matrix.IsHermitian.conjTranspose_eigenvectorMatrixInv
 
@@ -103,7 +103,7 @@ For the spectral theorem on linear maps, see
 theorem spectral_theorem :
     hA.eigenvectorMatrixInv ⬝ A = diagonal ((↑) ∘ hA.eigenvalues) ⬝ hA.eigenvectorMatrixInv := by
   rw [eigenvectorMatrixInv, PiLp.basis_toMatrix_basisFun_mul]
-  ext (i j)
+  ext i j
   have := isHermitian_iff_isSymmetric.1 hA
   convert this.eigenvectorBasis_apply_self_apply finrank_euclideanSpace (EuclideanSpace.single j 1)
     ((Fintype.equivOfCardEq (Fintype.card_fin _)).symm i) using 1

• depends on: #4920
• depends on: #5058