SesquilinearForm.lean

/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.LinearAlgebra.Basis.Bilinear

#align_import linear_algebra.matrix.sesquilinear_form from "leanprover-community/mathlib"@"84582d2872fb47c0c17eec7382dc097c9ec7137a"

/-!
# Sesquilinear form

This file defines the conversion between sesquilinear forms and matrices.

## Main definitions

 * `Matrix.toLinearMap₂` given a basis define a bilinear form
 * `Matrix.toLinearMap₂'` define the bilinear form on `n → R`
 * `LinearMap.toMatrix₂`: calculate the matrix coefficients of a bilinear form
 * `LinearMap.toMatrix₂'`: calculate the matrix coefficients of a bilinear form on `n → R`

## Todos

At the moment this is quite a literal port from `Matrix.BilinearForm`. Everything should be
generalized to fully semibilinear forms.

## Tags

sesquilinear_form, matrix, basis

-/


variable {R R₁ R₂ M M₁ M₂ M₁' M₂' n m n' m' ι : Type*}

open BigOperators

open Finset LinearMap Matrix

open Matrix

section AuxToLinearMap

variable [CommSemiring R] [Semiring R₁] [Semiring R₂]
variable [Fintype n] [Fintype m]
variable (σ₁ : R₁ →+* R) (σ₂ : R₂ →+* R)

/-- The map from `Matrix n n R` to bilinear forms on `n → R`.

This is an auxiliary definition for the equivalence `Matrix.toLinearMap₂'`. -/
def Matrix.toLinearMap₂'Aux (f : Matrix n m R) : (n → R₁) →ₛₗ[σ₁] (m → R₂) →ₛₗ[σ₂] R :=
  -- Porting note: we don't seem to have `∑ i j` as valid notation yet
  mk₂'ₛₗ σ₁ σ₂ (fun (v : n → R₁) (w : m → R₂) => ∑ i, ∑ j, σ₁ (v i) * f i j * σ₂ (w j))
    (fun _ _ _ => by simp only [Pi.add_apply, map_add, add_mul, sum_add_distrib])
    (fun _ _ _ => by simp only [Pi.smul_apply, smul_eq_mul, RingHom.map_mul, mul_assoc, mul_sum])
    (fun _ _ _ => by simp only [Pi.add_apply, map_add, mul_add, sum_add_distrib]) fun _ _ _ => by
    simp only [Pi.smul_apply, smul_eq_mul, RingHom.map_mul, mul_assoc, mul_left_comm, mul_sum]
#align matrix.to_linear_map₂'_aux Matrix.toLinearMap₂'Aux

variable [DecidableEq n] [DecidableEq m]

theorem Matrix.toLinearMap₂'Aux_stdBasis (f : Matrix n m R) (i : n) (j : m) :
    f.toLinearMap₂'Aux σ₁ σ₂ (LinearMap.stdBasis R₁ (fun _ => R₁) i 1)
      (LinearMap.stdBasis R₂ (fun _ => R₂) j 1) = f i j := by
  rw [Matrix.toLinearMap₂'Aux, mk₂'ₛₗ_apply]
  have : (∑ i', ∑ j', (if i = i' then 1 else 0) * f i' j' * if j = j' then 1 else 0) = f i j := by
    simp_rw [mul_assoc, ← Finset.mul_sum]
    simp only [boole_mul, Finset.sum_ite_eq, Finset.mem_univ, if_true, mul_comm (f _ _)]
  rw [← this]
  exact Finset.sum_congr rfl fun _ _ => Finset.sum_congr rfl fun _ _ => by simp
#align matrix.to_linear_map₂'_aux_std_basis Matrix.toLinearMap₂'Aux_stdBasis

end AuxToLinearMap

section AuxToMatrix

section CommSemiring

variable [CommSemiring R] [Semiring R₁] [Semiring R₂]
variable [AddCommMonoid M₁] [Module R₁ M₁] [AddCommMonoid M₂] [Module R₂ M₂]
variable {σ₁ : R₁ →+* R} {σ₂ : R₂ →+* R}

/-- The linear map from sesquilinear forms to `Matrix n m R` given an `n`-indexed basis for `M₁`
and an `m`-indexed basis for `M₂`.

This is an auxiliary definition for the equivalence `Matrix.toLinearMapₛₗ₂'`. -/
def LinearMap.toMatrix₂Aux (b₁ : n → M₁) (b₂ : m → M₂) :
    (M₁ →ₛₗ[σ₁] M₂ →ₛₗ[σ₂] R) →ₗ[R] Matrix n m R where
  toFun f := of fun i j => f (b₁ i) (b₂ j)
  map_add' _f _g := rfl
  map_smul' _f _g := rfl
#align linear_map.to_matrix₂_aux LinearMap.toMatrix₂Aux

@[simp]
theorem LinearMap.toMatrix₂Aux_apply (f : M₁ →ₛₗ[σ₁] M₂ →ₛₗ[σ₂] R) (b₁ : n → M₁) (b₂ : m → M₂)
    (i : n) (j : m) : LinearMap.toMatrix₂Aux b₁ b₂ f i j = f (b₁ i) (b₂ j) :=
  rfl
#align linear_map.to_matrix₂_aux_apply LinearMap.toMatrix₂Aux_apply

end CommSemiring

section CommRing

variable [CommSemiring R] [Semiring R₁] [Semiring R₂]
variable [AddCommMonoid M₁] [Module R₁ M₁] [AddCommMonoid M₂] [Module R₂ M₂]
variable [Fintype n] [Fintype m]
variable [DecidableEq n] [DecidableEq m]
variable {σ₁ : R₁ →+* R} {σ₂ : R₂ →+* R}

theorem LinearMap.toLinearMap₂'Aux_toMatrix₂Aux (f : (n → R₁) →ₛₗ[σ₁] (m → R₂) →ₛₗ[σ₂] R) :
    Matrix.toLinearMap₂'Aux σ₁ σ₂
        (LinearMap.toMatrix₂Aux (fun i => stdBasis R₁ (fun _ => R₁) i 1)
          (fun j => stdBasis R₂ (fun _ => R₂) j 1) f) =
      f := by
  refine' ext_basis (Pi.basisFun R₁ n) (Pi.basisFun R₂ m) fun i j => _
  simp_rw [Pi.basisFun_apply, Matrix.toLinearMap₂'Aux_stdBasis, LinearMap.toMatrix₂Aux_apply]
#align linear_map.to_linear_map₂'_aux_to_matrix₂_aux LinearMap.toLinearMap₂'Aux_toMatrix₂Aux

theorem Matrix.toMatrix₂Aux_toLinearMap₂'Aux (f : Matrix n m R) :
    LinearMap.toMatrix₂Aux (fun i => LinearMap.stdBasis R₁ (fun _ => R₁) i 1)
        (fun j => LinearMap.stdBasis R₂ (fun _ => R₂) j 1) (f.toLinearMap₂'Aux σ₁ σ₂) =
      f := by
  ext i j
  simp_rw [LinearMap.toMatrix₂Aux_apply, Matrix.toLinearMap₂'Aux_stdBasis]
#align matrix.to_matrix₂_aux_to_linear_map₂'_aux Matrix.toMatrix₂Aux_toLinearMap₂'Aux

end CommRing

end AuxToMatrix

section ToMatrix'

/-! ### Bilinear forms over `n → R`

This section deals with the conversion between matrices and sesquilinear forms on `n → R`.
-/


variable [CommSemiring R] [Semiring R₁] [Semiring R₂]
variable [Fintype n] [Fintype m]
variable [DecidableEq n] [DecidableEq m]
variable {σ₁ : R₁ →+* R} {σ₂ : R₂ →+* R}

/-- The linear equivalence between sesquilinear forms and `n × m` matrices -/
def LinearMap.toMatrixₛₗ₂' : ((n → R₁) →ₛₗ[σ₁] (m → R₂) →ₛₗ[σ₂] R) ≃ₗ[R] Matrix n m R :=
  {
    LinearMap.toMatrix₂Aux (fun i => stdBasis R₁ (fun _ => R₁) i 1) fun j =>
      stdBasis R₂ (fun _ => R₂) j
        1 with
    toFun := LinearMap.toMatrix₂Aux _ _
    invFun := Matrix.toLinearMap₂'Aux σ₁ σ₂
    left_inv := LinearMap.toLinearMap₂'Aux_toMatrix₂Aux
    right_inv := Matrix.toMatrix₂Aux_toLinearMap₂'Aux }
#align linear_map.to_matrixₛₗ₂' LinearMap.toMatrixₛₗ₂'

/-- The linear equivalence between bilinear forms and `n × m` matrices -/
def LinearMap.toMatrix₂' : ((n → R) →ₗ[R] (m → R) →ₗ[R] R) ≃ₗ[R] Matrix n m R :=
  LinearMap.toMatrixₛₗ₂'
#align linear_map.to_matrix₂' LinearMap.toMatrix₂'

variable (σ₁ σ₂)

/-- The linear equivalence between `n × n` matrices and sesquilinear forms on `n → R` -/
def Matrix.toLinearMapₛₗ₂' : Matrix n m R ≃ₗ[R] (n → R₁) →ₛₗ[σ₁] (m → R₂) →ₛₗ[σ₂] R :=
  LinearMap.toMatrixₛₗ₂'.symm
#align matrix.to_linear_mapₛₗ₂' Matrix.toLinearMapₛₗ₂'

/-- The linear equivalence between `n × n` matrices and bilinear forms on `n → R` -/
def Matrix.toLinearMap₂' : Matrix n m R ≃ₗ[R] (n → R) →ₗ[R] (m → R) →ₗ[R] R :=
  LinearMap.toMatrix₂'.symm
#align matrix.to_linear_map₂' Matrix.toLinearMap₂'

theorem Matrix.toLinearMapₛₗ₂'_aux_eq (M : Matrix n m R) :
    Matrix.toLinearMap₂'Aux σ₁ σ₂ M = Matrix.toLinearMapₛₗ₂' σ₁ σ₂ M :=
  rfl
#align matrix.to_linear_mapₛₗ₂'_aux_eq Matrix.toLinearMapₛₗ₂'_aux_eq

theorem Matrix.toLinearMapₛₗ₂'_apply (M : Matrix n m R) (x : n → R₁) (y : m → R₂) :
    -- Porting note: we don't seem to have `∑ i j` as valid notation yet
    Matrix.toLinearMapₛₗ₂' σ₁ σ₂ M x y = ∑ i, ∑ j, σ₁ (x i) * M i j * σ₂ (y j) :=
  rfl
#align matrix.to_linear_mapₛₗ₂'_apply Matrix.toLinearMapₛₗ₂'_apply

theorem Matrix.toLinearMap₂'_apply (M : Matrix n m R) (x : n → R) (y : m → R) :
    -- Porting note: we don't seem to have `∑ i j` as valid notation yet
    Matrix.toLinearMap₂' M x y = ∑ i, ∑ j, x i * M i j * y j :=
  rfl
#align matrix.to_linear_map₂'_apply Matrix.toLinearMap₂'_apply

theorem Matrix.toLinearMap₂'_apply' (M : Matrix n m R) (v : n → R) (w : m → R) :
    Matrix.toLinearMap₂' M v w = Matrix.dotProduct v (M *ᵥ w) := by
  simp_rw [Matrix.toLinearMap₂'_apply, Matrix.dotProduct, Matrix.mulVec, Matrix.dotProduct]
  refine' Finset.sum_congr rfl fun _ _ => _
  rw [Finset.mul_sum]
  refine' Finset.sum_congr rfl fun _ _ => _
  rw [← mul_assoc]
#align matrix.to_linear_map₂'_apply' Matrix.toLinearMap₂'_apply'

@[simp]
theorem Matrix.toLinearMapₛₗ₂'_stdBasis (M : Matrix n m R) (i : n) (j : m) :
    Matrix.toLinearMapₛₗ₂' σ₁ σ₂ M (LinearMap.stdBasis R₁ (fun _ => R₁) i 1)
      (LinearMap.stdBasis R₂ (fun _ => R₂) j 1) = M i j :=
  Matrix.toLinearMap₂'Aux_stdBasis σ₁ σ₂ M i j
#align matrix.to_linear_mapₛₗ₂'_std_basis Matrix.toLinearMapₛₗ₂'_stdBasis

@[simp]
theorem Matrix.toLinearMap₂'_stdBasis (M : Matrix n m R) (i : n) (j : m) :
    Matrix.toLinearMap₂' M (LinearMap.stdBasis R (fun _ => R) i 1)
      (LinearMap.stdBasis R (fun _ => R) j 1) = M i j :=
  Matrix.toLinearMap₂'Aux_stdBasis _ _ M i j
#align matrix.to_linear_map₂'_std_basis Matrix.toLinearMap₂'_stdBasis

@[simp]
theorem LinearMap.toMatrixₛₗ₂'_symm :
    (LinearMap.toMatrixₛₗ₂'.symm : Matrix n m R ≃ₗ[R] _) = Matrix.toLinearMapₛₗ₂' σ₁ σ₂ :=
  rfl
#align linear_map.to_matrixₛₗ₂'_symm LinearMap.toMatrixₛₗ₂'_symm

@[simp]
theorem Matrix.toLinearMapₛₗ₂'_symm :
    ((Matrix.toLinearMapₛₗ₂' σ₁ σ₂).symm : _ ≃ₗ[R] Matrix n m R) = LinearMap.toMatrixₛₗ₂' :=
  LinearMap.toMatrixₛₗ₂'.symm_symm
#align matrix.to_linear_mapₛₗ₂'_symm Matrix.toLinearMapₛₗ₂'_symm

@[simp]
theorem Matrix.toLinearMapₛₗ₂'_toMatrix' (B : (n → R₁) →ₛₗ[σ₁] (m → R₂) →ₛₗ[σ₂] R) :
    Matrix.toLinearMapₛₗ₂' σ₁ σ₂ (LinearMap.toMatrixₛₗ₂' B) = B :=
  (Matrix.toLinearMapₛₗ₂' σ₁ σ₂).apply_symm_apply B
#align matrix.to_linear_mapₛₗ₂'_to_matrix' Matrix.toLinearMapₛₗ₂'_toMatrix'

@[simp]
theorem Matrix.toLinearMap₂'_toMatrix' (B : (n → R) →ₗ[R] (m → R) →ₗ[R] R) :
    Matrix.toLinearMap₂' (LinearMap.toMatrix₂' B) = B :=
  Matrix.toLinearMap₂'.apply_symm_apply B
#align matrix.to_linear_map₂'_to_matrix' Matrix.toLinearMap₂'_toMatrix'

@[simp]
theorem LinearMap.toMatrix'_toLinearMapₛₗ₂' (M : Matrix n m R) :
    LinearMap.toMatrixₛₗ₂' (Matrix.toLinearMapₛₗ₂' σ₁ σ₂ M) = M :=
  LinearMap.toMatrixₛₗ₂'.apply_symm_apply M
#align linear_map.to_matrix'_to_linear_mapₛₗ₂' LinearMap.toMatrix'_toLinearMapₛₗ₂'

@[simp]
theorem LinearMap.toMatrix'_toLinearMap₂' (M : Matrix n m R) :
    LinearMap.toMatrix₂' (Matrix.toLinearMap₂' M) = M :=
  LinearMap.toMatrixₛₗ₂'.apply_symm_apply M
#align linear_map.to_matrix'_to_linear_map₂' LinearMap.toMatrix'_toLinearMap₂'

@[simp]
theorem LinearMap.toMatrixₛₗ₂'_apply (B : (n → R₁) →ₛₗ[σ₁] (m → R₂) →ₛₗ[σ₂] R) (i : n) (j : m) :
    LinearMap.toMatrixₛₗ₂' B i j =
      B (stdBasis R₁ (fun _ => R₁) i 1) (stdBasis R₂ (fun _ => R₂) j 1) :=
  rfl
#align linear_map.to_matrixₛₗ₂'_apply LinearMap.toMatrixₛₗ₂'_apply

@[simp]
theorem LinearMap.toMatrix₂'_apply (B : (n → R) →ₗ[R] (m → R) →ₗ[R] R) (i : n) (j : m) :
    LinearMap.toMatrix₂' B i j = B (stdBasis R (fun _ => R) i 1) (stdBasis R (fun _ => R) j 1) :=
  rfl
#align linear_map.to_matrix₂'_apply LinearMap.toMatrix₂'_apply

variable [Fintype n'] [Fintype m']
variable [DecidableEq n'] [DecidableEq m']

@[simp]
theorem LinearMap.toMatrix₂'_compl₁₂ (B : (n → R) →ₗ[R] (m → R) →ₗ[R] R) (l : (n' → R) →ₗ[R] n → R)
    (r : (m' → R) →ₗ[R] m → R) :
    toMatrix₂' (B.compl₁₂ l r) = (toMatrix' l)ᵀ * toMatrix₂' B * toMatrix' r := by
  ext i j
  simp only [LinearMap.toMatrix₂'_apply, LinearMap.compl₁₂_apply, transpose_apply, Matrix.mul_apply,
    LinearMap.toMatrix', LinearEquiv.coe_mk, sum_mul]
  rw [sum_comm]
  conv_lhs => rw [← LinearMap.sum_repr_mul_repr_mul (Pi.basisFun R n) (Pi.basisFun R m) (l _) (r _)]
  rw [Finsupp.sum_fintype]
  · apply sum_congr rfl
    rintro i' -
    rw [Finsupp.sum_fintype]
    · apply sum_congr rfl
      rintro j' -
      simp only [smul_eq_mul, Pi.basisFun_repr, mul_assoc, mul_comm, mul_left_comm,
        Pi.basisFun_apply, of_apply]
    · intros
      simp only [zero_smul, smul_zero]
  · intros
    simp only [zero_smul, Finsupp.sum_zero]
#align linear_map.to_matrix₂'_compl₁₂ LinearMap.toMatrix₂'_compl₁₂

theorem LinearMap.toMatrix₂'_comp (B : (n → R) →ₗ[R] (m → R) →ₗ[R] R) (f : (n' → R) →ₗ[R] n → R) :
    toMatrix₂' (B.comp f) = (toMatrix' f)ᵀ * toMatrix₂' B := by
  rw [← LinearMap.compl₂_id (B.comp f), ← LinearMap.compl₁₂]
  simp
#align linear_map.to_matrix₂'_comp LinearMap.toMatrix₂'_comp

theorem LinearMap.toMatrix₂'_compl₂ (B : (n → R) →ₗ[R] (m → R) →ₗ[R] R) (f : (m' → R) →ₗ[R] m → R) :
    toMatrix₂' (B.compl₂ f) = toMatrix₂' B * toMatrix' f := by
  rw [← LinearMap.comp_id B, ← LinearMap.compl₁₂]
  simp
#align linear_map.to_matrix₂'_compl₂ LinearMap.toMatrix₂'_compl₂

theorem LinearMap.mul_toMatrix₂'_mul (B : (n → R) →ₗ[R] (m → R) →ₗ[R] R) (M : Matrix n' n R)
    (N : Matrix m m' R) : M * toMatrix₂' B * N = toMatrix₂' (B.compl₁₂ (toLin' Mᵀ) (toLin' N)) := by
  simp
#align linear_map.mul_to_matrix₂'_mul LinearMap.mul_toMatrix₂'_mul

theorem LinearMap.mul_toMatrix' (B : (n → R) →ₗ[R] (m → R) →ₗ[R] R) (M : Matrix n' n R) :
    M * toMatrix₂' B = toMatrix₂' (B.comp <| toLin' Mᵀ) := by
  simp only [B.toMatrix₂'_comp, transpose_transpose, toMatrix'_toLin']
#align linear_map.mul_to_matrix' LinearMap.mul_toMatrix'

theorem LinearMap.toMatrix₂'_mul (B : (n → R) →ₗ[R] (m → R) →ₗ[R] R) (M : Matrix m m' R) :
    toMatrix₂' B * M = toMatrix₂' (B.compl₂ <| toLin' M) := by
  simp only [B.toMatrix₂'_compl₂, toMatrix'_toLin']
#align linear_map.to_matrix₂'_mul LinearMap.toMatrix₂'_mul

theorem Matrix.toLinearMap₂'_comp (M : Matrix n m R) (P : Matrix n n' R) (Q : Matrix m m' R) :
    M.toLinearMap₂'.compl₁₂ (toLin' P) (toLin' Q) = toLinearMap₂' (Pᵀ * M * Q) :=
  LinearMap.toMatrix₂'.injective (by simp)
#align matrix.to_linear_map₂'_comp Matrix.toLinearMap₂'_comp

end ToMatrix'

section ToMatrix

/-! ### Bilinear forms over arbitrary vector spaces

This section deals with the conversion between matrices and bilinear forms on
a module with a fixed basis.
-/


variable [CommSemiring R]
variable [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂]
variable [DecidableEq n] [Fintype n]
variable [DecidableEq m] [Fintype m]
variable (b₁ : Basis n R M₁) (b₂ : Basis m R M₂)

/-- `LinearMap.toMatrix₂ b₁ b₂` is the equivalence between `R`-bilinear forms on `M` and
`n`-by-`m` matrices with entries in `R`, if `b₁` and `b₂` are `R`-bases for `M₁` and `M₂`,
respectively. -/
noncomputable def LinearMap.toMatrix₂ : (M₁ →ₗ[R] M₂ →ₗ[R] R) ≃ₗ[R] Matrix n m R :=
  (b₁.equivFun.arrowCongr (b₂.equivFun.arrowCongr (LinearEquiv.refl R R))).trans
    LinearMap.toMatrix₂'
#align linear_map.to_matrix₂ LinearMap.toMatrix₂

/-- `Matrix.toLinearMap₂ b₁ b₂` is the equivalence between `R`-bilinear forms on `M` and
`n`-by-`m` matrices with entries in `R`, if `b₁` and `b₂` are `R`-bases for `M₁` and `M₂`,
respectively; this is the reverse direction of `LinearMap.toMatrix₂ b₁ b₂`. -/
noncomputable def Matrix.toLinearMap₂ : Matrix n m R ≃ₗ[R] M₁ →ₗ[R] M₂ →ₗ[R] R :=
  (LinearMap.toMatrix₂ b₁ b₂).symm
#align matrix.to_linear_map₂ Matrix.toLinearMap₂

-- We make this and not `LinearMap.toMatrix₂` a `simp` lemma to avoid timeouts
@[simp]
theorem LinearMap.toMatrix₂_apply (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (i : n) (j : m) :
    LinearMap.toMatrix₂ b₁ b₂ B i j = B (b₁ i) (b₂ j) := by
  simp only [LinearMap.toMatrix₂, LinearEquiv.trans_apply, LinearMap.toMatrix₂'_apply,
    LinearEquiv.trans_apply, LinearMap.toMatrix₂'_apply, LinearEquiv.arrowCongr_apply,
    Basis.equivFun_symm_stdBasis, LinearEquiv.refl_apply]
#align linear_map.to_matrix₂_apply LinearMap.toMatrix₂_apply

@[simp]
theorem Matrix.toLinearMap₂_apply (M : Matrix n m R) (x : M₁) (y : M₂) :
    Matrix.toLinearMap₂ b₁ b₂ M x y = ∑ i, ∑ j, b₁.repr x i * M i j * b₂.repr y j :=
  rfl
#align matrix.to_linear_map₂_apply Matrix.toLinearMap₂_apply

-- Not a `simp` lemma since `LinearMap.toMatrix₂` needs an extra argument
theorem LinearMap.toMatrix₂Aux_eq (B : M₁ →ₗ[R] M₂ →ₗ[R] R) :
    LinearMap.toMatrix₂Aux b₁ b₂ B = LinearMap.toMatrix₂ b₁ b₂ B :=
  Matrix.ext fun i j => by rw [LinearMap.toMatrix₂_apply, LinearMap.toMatrix₂Aux_apply]
#align linear_map.to_matrix₂_aux_eq LinearMap.toMatrix₂Aux_eq

@[simp]
theorem LinearMap.toMatrix₂_symm : (LinearMap.toMatrix₂ b₁ b₂).symm = Matrix.toLinearMap₂ b₁ b₂ :=
  rfl
#align linear_map.to_matrix₂_symm LinearMap.toMatrix₂_symm

@[simp]
theorem Matrix.toLinearMap₂_symm : (Matrix.toLinearMap₂ b₁ b₂).symm = LinearMap.toMatrix₂ b₁ b₂ :=
  (LinearMap.toMatrix₂ b₁ b₂).symm_symm
#align matrix.to_linear_map₂_symm Matrix.toLinearMap₂_symm

theorem Matrix.toLinearMap₂_basisFun :
    Matrix.toLinearMap₂ (Pi.basisFun R n) (Pi.basisFun R m) = Matrix.toLinearMap₂' := by
  ext M
  simp only [Matrix.toLinearMap₂_apply, Matrix.toLinearMap₂'_apply, Pi.basisFun_repr, coe_comp,
    Function.comp_apply]
#align matrix.to_linear_map₂_basis_fun Matrix.toLinearMap₂_basisFun

theorem LinearMap.toMatrix₂_basisFun :
    LinearMap.toMatrix₂ (Pi.basisFun R n) (Pi.basisFun R m) = LinearMap.toMatrix₂' := by
  ext B
  rw [LinearMap.toMatrix₂_apply, LinearMap.toMatrix₂'_apply, Pi.basisFun_apply, Pi.basisFun_apply]
#align linear_map.to_matrix₂_basis_fun LinearMap.toMatrix₂_basisFun

@[simp]
theorem Matrix.toLinearMap₂_toMatrix₂ (B : M₁ →ₗ[R] M₂ →ₗ[R] R) :
    Matrix.toLinearMap₂ b₁ b₂ (LinearMap.toMatrix₂ b₁ b₂ B) = B :=
  (Matrix.toLinearMap₂ b₁ b₂).apply_symm_apply B
#align matrix.to_linear_map₂_to_matrix₂ Matrix.toLinearMap₂_toMatrix₂

@[simp]
theorem LinearMap.toMatrix₂_toLinearMap₂ (M : Matrix n m R) :
    LinearMap.toMatrix₂ b₁ b₂ (Matrix.toLinearMap₂ b₁ b₂ M) = M :=
  (LinearMap.toMatrix₂ b₁ b₂).apply_symm_apply M
#align linear_map.to_matrix₂_to_linear_map₂ LinearMap.toMatrix₂_toLinearMap₂

variable [AddCommMonoid M₁'] [Module R M₁']
variable [AddCommMonoid M₂'] [Module R M₂']
variable (b₁' : Basis n' R M₁')
variable (b₂' : Basis m' R M₂')
variable [Fintype n'] [Fintype m']
variable [DecidableEq n'] [DecidableEq m']

-- Cannot be a `simp` lemma because `b₁` and `b₂` must be inferred.
theorem LinearMap.toMatrix₂_compl₁₂ (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (l : M₁' →ₗ[R] M₁)
    (r : M₂' →ₗ[R] M₂) :
    LinearMap.toMatrix₂ b₁' b₂' (B.compl₁₂ l r) =
      (toMatrix b₁' b₁ l)ᵀ * LinearMap.toMatrix₂ b₁ b₂ B * toMatrix b₂' b₂ r := by
  ext i j
  simp only [LinearMap.toMatrix₂_apply, compl₁₂_apply, transpose_apply, Matrix.mul_apply,
    LinearMap.toMatrix_apply, LinearEquiv.coe_mk, sum_mul]
  rw [sum_comm]
  conv_lhs => rw [← LinearMap.sum_repr_mul_repr_mul b₁ b₂]
  rw [Finsupp.sum_fintype]
  · apply sum_congr rfl
    rintro i' -
    rw [Finsupp.sum_fintype]
    · apply sum_congr rfl
      rintro j' -
      simp only [smul_eq_mul, LinearMap.toMatrix_apply, Basis.equivFun_apply, mul_assoc, mul_comm,
        mul_left_comm]
    · intros
      simp only [zero_smul, smul_zero]
  · intros
    simp only [zero_smul, Finsupp.sum_zero]
#align linear_map.to_matrix₂_compl₁₂ LinearMap.toMatrix₂_compl₁₂

theorem LinearMap.toMatrix₂_comp (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (f : M₁' →ₗ[R] M₁) :
    LinearMap.toMatrix₂ b₁' b₂ (B.comp f) = (toMatrix b₁' b₁ f)ᵀ * LinearMap.toMatrix₂ b₁ b₂ B := by
  rw [← LinearMap.compl₂_id (B.comp f), ← LinearMap.compl₁₂, LinearMap.toMatrix₂_compl₁₂ b₁ b₂]
  simp
#align linear_map.to_matrix₂_comp LinearMap.toMatrix₂_comp

theorem LinearMap.toMatrix₂_compl₂ (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (f : M₂' →ₗ[R] M₂) :
    LinearMap.toMatrix₂ b₁ b₂' (B.compl₂ f) = LinearMap.toMatrix₂ b₁ b₂ B * toMatrix b₂' b₂ f := by
  rw [← LinearMap.comp_id B, ← LinearMap.compl₁₂, LinearMap.toMatrix₂_compl₁₂ b₁ b₂]
  simp
#align linear_map.to_matrix₂_compl₂ LinearMap.toMatrix₂_compl₂

@[simp]
theorem LinearMap.toMatrix₂_mul_basis_toMatrix (c₁ : Basis n' R M₁) (c₂ : Basis m' R M₂)
    (B : M₁ →ₗ[R] M₂ →ₗ[R] R) :
    (b₁.toMatrix c₁)ᵀ * LinearMap.toMatrix₂ b₁ b₂ B * b₂.toMatrix c₂ =
      LinearMap.toMatrix₂ c₁ c₂ B := by
  simp_rw [← LinearMap.toMatrix_id_eq_basis_toMatrix]
  rw [← LinearMap.toMatrix₂_compl₁₂, LinearMap.compl₁₂_id_id]
#align linear_map.to_matrix₂_mul_basis_to_matrix LinearMap.toMatrix₂_mul_basis_toMatrix

theorem LinearMap.mul_toMatrix₂_mul (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (M : Matrix n' n R)
    (N : Matrix m m' R) :
    M * LinearMap.toMatrix₂ b₁ b₂ B * N =
      LinearMap.toMatrix₂ b₁' b₂' (B.compl₁₂ (toLin b₁' b₁ Mᵀ) (toLin b₂' b₂ N)) :=
  by simp_rw [LinearMap.toMatrix₂_compl₁₂ b₁ b₂, toMatrix_toLin, transpose_transpose]
#align linear_map.mul_to_matrix₂_mul LinearMap.mul_toMatrix₂_mul

theorem LinearMap.mul_toMatrix₂ (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (M : Matrix n' n R) :
    M * LinearMap.toMatrix₂ b₁ b₂ B = LinearMap.toMatrix₂ b₁' b₂ (B.comp (toLin b₁' b₁ Mᵀ)) := by
  rw [LinearMap.toMatrix₂_comp b₁, toMatrix_toLin, transpose_transpose]
#align linear_map.mul_to_matrix₂ LinearMap.mul_toMatrix₂

theorem LinearMap.toMatrix₂_mul (B : M₁ →ₗ[R] M₂ →ₗ[R] R) (M : Matrix m m' R) :
    LinearMap.toMatrix₂ b₁ b₂ B * M = LinearMap.toMatrix₂ b₁ b₂' (B.compl₂ (toLin b₂' b₂ M)) := by
  rw [LinearMap.toMatrix₂_compl₂ b₁ b₂, toMatrix_toLin]
#align linear_map.to_matrix₂_mul LinearMap.toMatrix₂_mul

theorem Matrix.toLinearMap₂_compl₁₂ (M : Matrix n m R) (P : Matrix n n' R) (Q : Matrix m m' R) :
    (Matrix.toLinearMap₂ b₁ b₂ M).compl₁₂ (toLin b₁' b₁ P) (toLin b₂' b₂ Q) =
      Matrix.toLinearMap₂ b₁' b₂' (Pᵀ * M * Q) :=
  (LinearMap.toMatrix₂ b₁' b₂').injective
    (by
      simp only [LinearMap.toMatrix₂_compl₁₂ b₁ b₂, LinearMap.toMatrix₂_toLinearMap₂,
        toMatrix_toLin])
#align matrix.to_linear_map₂_compl₁₂ Matrix.toLinearMap₂_compl₁₂

end ToMatrix

/-! ### Adjoint pairs-/


section MatrixAdjoints

open Matrix

variable [CommRing R]
variable [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂]
variable [Fintype n] [Fintype n']
variable (b₁ : Basis n R M₁) (b₂ : Basis n' R M₂)
variable (J J₂ : Matrix n n R) (J' : Matrix n' n' R)
variable (A : Matrix n' n R) (A' : Matrix n n' R)
variable (A₁ A₂ : Matrix n n R)

/-- The condition for the matrices `A`, `A'` to be an adjoint pair with respect to the square
matrices `J`, `J₃`. -/
def Matrix.IsAdjointPair :=
  Aᵀ * J' = J * A'
#align matrix.is_adjoint_pair Matrix.IsAdjointPair

/-- The condition for a square matrix `A` to be self-adjoint with respect to the square matrix
`J`. -/
def Matrix.IsSelfAdjoint :=
  Matrix.IsAdjointPair J J A₁ A₁
#align matrix.is_self_adjoint Matrix.IsSelfAdjoint

/-- The condition for a square matrix `A` to be skew-adjoint with respect to the square matrix
`J`. -/
def Matrix.IsSkewAdjoint :=
  Matrix.IsAdjointPair J J A₁ (-A₁)
#align matrix.is_skew_adjoint Matrix.IsSkewAdjoint

variable [DecidableEq n] [DecidableEq n']

@[simp]
theorem isAdjointPair_toLinearMap₂' :
    LinearMap.IsAdjointPair (Matrix.toLinearMap₂' J) (Matrix.toLinearMap₂' J') (Matrix.toLin' A)
        (Matrix.toLin' A') ↔
      Matrix.IsAdjointPair J J' A A' := by
  rw [isAdjointPair_iff_comp_eq_compl₂]
  have h :
    ∀ B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R,
      B = B' ↔ LinearMap.toMatrix₂' B = LinearMap.toMatrix₂' B' := by
    intro B B'
    constructor <;> intro h
    · rw [h]
    · exact LinearMap.toMatrix₂'.injective h
  simp_rw [h, LinearMap.toMatrix₂'_comp, LinearMap.toMatrix₂'_compl₂, LinearMap.toMatrix'_toLin',
    LinearMap.toMatrix'_toLinearMap₂']
  rfl
#align is_adjoint_pair_to_linear_map₂' isAdjointPair_toLinearMap₂'

@[simp]
theorem isAdjointPair_toLinearMap₂ :
    LinearMap.IsAdjointPair (Matrix.toLinearMap₂ b₁ b₁ J) (Matrix.toLinearMap₂ b₂ b₂ J')
        (Matrix.toLin b₁ b₂ A) (Matrix.toLin b₂ b₁ A') ↔
      Matrix.IsAdjointPair J J' A A' := by
  rw [isAdjointPair_iff_comp_eq_compl₂]
  have h :
    ∀ B B' : M₁ →ₗ[R] M₂ →ₗ[R] R,
      B = B' ↔ LinearMap.toMatrix₂ b₁ b₂ B = LinearMap.toMatrix₂ b₁ b₂ B' := by
    intro B B'
    constructor <;> intro h
    · rw [h]
    · exact (LinearMap.toMatrix₂ b₁ b₂).injective h
  simp_rw [h, LinearMap.toMatrix₂_comp b₂ b₂, LinearMap.toMatrix₂_compl₂ b₁ b₁,
    LinearMap.toMatrix_toLin, LinearMap.toMatrix₂_toLinearMap₂]
  rfl
#align is_adjoint_pair_to_linear_map₂ isAdjointPair_toLinearMap₂

theorem Matrix.isAdjointPair_equiv (P : Matrix n n R) (h : IsUnit P) :
    (Pᵀ * J * P).IsAdjointPair (Pᵀ * J * P) A₁ A₂ ↔
      J.IsAdjointPair J (P * A₁ * P⁻¹) (P * A₂ * P⁻¹) := by
  have h' : IsUnit P.det := P.isUnit_iff_isUnit_det.mp h
  let u := P.nonsingInvUnit h'
  let v := Pᵀ.nonsingInvUnit (P.isUnit_det_transpose h')
  let x := A₁ᵀ * Pᵀ * J
  let y := J * P * A₂
  -- TODO(mathlib4#6607): fix elaboration so `val` isn't needed
  suffices x * u.val = v.val * y ↔ (v⁻¹).val * x = y * (u⁻¹).val by
    dsimp only [Matrix.IsAdjointPair]
    simp only [Matrix.transpose_mul]
    simp only [← mul_assoc, P.transpose_nonsing_inv]
    -- Porting note: the previous proof used `conv` and was causing timeouts, so we use `convert`
    convert this using 2
    · rw [mul_assoc, mul_assoc, ← mul_assoc J]
      rfl
    · rw [mul_assoc, mul_assoc, ← mul_assoc _ _ J]
      rfl
  rw [Units.eq_mul_inv_iff_mul_eq]
  conv_rhs => rw [mul_assoc]
  rw [v.inv_mul_eq_iff_eq_mul]
#align matrix.is_adjoint_pair_equiv Matrix.isAdjointPair_equiv

/-- The submodule of pair-self-adjoint matrices with respect to bilinear forms corresponding to
given matrices `J`, `J₂`. -/
def pairSelfAdjointMatricesSubmodule : Submodule R (Matrix n n R) :=
  (isPairSelfAdjointSubmodule (Matrix.toLinearMap₂' J) (Matrix.toLinearMap₂' J₂)).map
    ((LinearMap.toMatrix' : ((n → R) →ₗ[R] n → R) ≃ₗ[R] Matrix n n R) :
      ((n → R) →ₗ[R] n → R) →ₗ[R] Matrix n n R)
#align pair_self_adjoint_matrices_submodule pairSelfAdjointMatricesSubmodule

@[simp]
theorem mem_pairSelfAdjointMatricesSubmodule :
    A₁ ∈ pairSelfAdjointMatricesSubmodule J J₂ ↔ Matrix.IsAdjointPair J J₂ A₁ A₁ := by
  simp only [pairSelfAdjointMatricesSubmodule, LinearEquiv.coe_coe, LinearMap.toMatrix'_apply,
    Submodule.mem_map, mem_isPairSelfAdjointSubmodule]
  constructor
  · rintro ⟨f, hf, hA⟩
    have hf' : f = toLin' A₁ := by rw [← hA, Matrix.toLin'_toMatrix']
    rw [hf'] at hf
    rw [← isAdjointPair_toLinearMap₂']
    exact hf
  · intro h
    refine' ⟨toLin' A₁, _, LinearMap.toMatrix'_toLin' _⟩
    exact (isAdjointPair_toLinearMap₂' _ _ _ _).mpr h
#align mem_pair_self_adjoint_matrices_submodule mem_pairSelfAdjointMatricesSubmodule

/-- The submodule of self-adjoint matrices with respect to the bilinear form corresponding to
the matrix `J`. -/
def selfAdjointMatricesSubmodule : Submodule R (Matrix n n R) :=
  pairSelfAdjointMatricesSubmodule J J
#align self_adjoint_matrices_submodule selfAdjointMatricesSubmodule

@[simp]
theorem mem_selfAdjointMatricesSubmodule :
    A₁ ∈ selfAdjointMatricesSubmodule J ↔ J.IsSelfAdjoint A₁ := by
  erw [mem_pairSelfAdjointMatricesSubmodule]
  rfl
#align mem_self_adjoint_matrices_submodule mem_selfAdjointMatricesSubmodule

/-- The submodule of skew-adjoint matrices with respect to the bilinear form corresponding to
the matrix `J`. -/
def skewAdjointMatricesSubmodule : Submodule R (Matrix n n R) :=
  pairSelfAdjointMatricesSubmodule (-J) J
#align skew_adjoint_matrices_submodule skewAdjointMatricesSubmodule

@[simp]
theorem mem_skewAdjointMatricesSubmodule :
    A₁ ∈ skewAdjointMatricesSubmodule J ↔ J.IsSkewAdjoint A₁ := by
  erw [mem_pairSelfAdjointMatricesSubmodule]
  simp [Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]
#align mem_skew_adjoint_matrices_submodule mem_skewAdjointMatricesSubmodule

end MatrixAdjoints

namespace LinearMap

/-! ### Nondegenerate bilinear forms-/


section Det

open Matrix

variable [CommRing R₁] [AddCommMonoid M₁] [Module R₁ M₁]
variable [DecidableEq ι] [Fintype ι]

theorem _root_.Matrix.separatingLeft_toLinearMap₂'_iff_separatingLeft_toLinearMap₂
    {M : Matrix ι ι R₁} (b : Basis ι R₁ M₁) :
    M.toLinearMap₂'.SeparatingLeft ↔ (Matrix.toLinearMap₂ b b M).SeparatingLeft :=
  (separatingLeft_congr_iff b.equivFun.symm b.equivFun.symm).symm
#align matrix.separating_left_to_linear_map₂'_iff_separating_left_to_linear_map₂ Matrix.separatingLeft_toLinearMap₂'_iff_separatingLeft_toLinearMap₂

variable (B : M₁ →ₗ[R₁] M₁ →ₗ[R₁] R₁)

-- Lemmas transferring nondegeneracy between a matrix and its associated bilinear form
theorem _root_.Matrix.Nondegenerate.toLinearMap₂' {M : Matrix ι ι R₁} (h : M.Nondegenerate) :
    M.toLinearMap₂'.SeparatingLeft := fun x hx =>
  h.eq_zero_of_ortho fun y => by simpa only [toLinearMap₂'_apply'] using hx y
#align matrix.nondegenerate.to_linear_map₂' Matrix.Nondegenerate.toLinearMap₂'

@[simp]
theorem _root_.Matrix.separatingLeft_toLinearMap₂'_iff {M : Matrix ι ι R₁} :
    M.toLinearMap₂'.SeparatingLeft ↔ M.Nondegenerate :=
  ⟨fun h v hv => h v fun w => (M.toLinearMap₂'_apply' _ _).trans <| hv w,
    Matrix.Nondegenerate.toLinearMap₂'⟩
#align matrix.separating_left_to_linear_map₂'_iff Matrix.separatingLeft_toLinearMap₂'_iff

theorem _root_.Matrix.Nondegenerate.toLinearMap₂ {M : Matrix ι ι R₁} (h : M.Nondegenerate)
    (b : Basis ι R₁ M₁) : (toLinearMap₂ b b M).SeparatingLeft :=
  (Matrix.separatingLeft_toLinearMap₂'_iff_separatingLeft_toLinearMap₂ b).mp h.toLinearMap₂'
#align matrix.nondegenerate.to_linear_map₂ Matrix.Nondegenerate.toLinearMap₂

@[simp]
theorem _root_.Matrix.separatingLeft_toLinearMap₂_iff {M : Matrix ι ι R₁} (b : Basis ι R₁ M₁) :
    (toLinearMap₂ b b M).SeparatingLeft ↔ M.Nondegenerate := by
  rw [← Matrix.separatingLeft_toLinearMap₂'_iff_separatingLeft_toLinearMap₂,
    Matrix.separatingLeft_toLinearMap₂'_iff]
#align matrix.separating_left_to_linear_map₂_iff Matrix.separatingLeft_toLinearMap₂_iff

-- Lemmas transferring nondegeneracy between a bilinear form and its associated matrix
@[simp]
theorem nondegenerate_toMatrix₂'_iff {B : (ι → R₁) →ₗ[R₁] (ι → R₁) →ₗ[R₁] R₁} :
    B.toMatrix₂'.Nondegenerate ↔ B.SeparatingLeft :=
  Matrix.separatingLeft_toLinearMap₂'_iff.symm.trans <|
    (Matrix.toLinearMap₂'_toMatrix' B).symm ▸ Iff.rfl
#align linear_map.nondegenerate_to_matrix₂'_iff LinearMap.nondegenerate_toMatrix₂'_iff

theorem SeparatingLeft.toMatrix₂' {B : (ι → R₁) →ₗ[R₁] (ι → R₁) →ₗ[R₁] R₁} (h : B.SeparatingLeft) :
    B.toMatrix₂'.Nondegenerate :=
  nondegenerate_toMatrix₂'_iff.mpr h
#align linear_map.separating_left.to_matrix₂' LinearMap.SeparatingLeft.toMatrix₂'

@[simp]
theorem nondegenerate_toMatrix_iff {B : M₁ →ₗ[R₁] M₁ →ₗ[R₁] R₁} (b : Basis ι R₁ M₁) :
    (toMatrix₂ b b B).Nondegenerate ↔ B.SeparatingLeft :=
  (Matrix.separatingLeft_toLinearMap₂_iff b).symm.trans <|
    (Matrix.toLinearMap₂_toMatrix₂ b b B).symm ▸ Iff.rfl
#align linear_map.nondegenerate_to_matrix_iff LinearMap.nondegenerate_toMatrix_iff

theorem SeparatingLeft.toMatrix₂ {B : M₁ →ₗ[R₁] M₁ →ₗ[R₁] R₁} (h : B.SeparatingLeft)
    (b : Basis ι R₁ M₁) : (toMatrix₂ b b B).Nondegenerate :=
  (nondegenerate_toMatrix_iff b).mpr h
#align linear_map.separating_left.to_matrix₂ LinearMap.SeparatingLeft.toMatrix₂

-- Some shorthands for combining the above with `Matrix.nondegenerate_of_det_ne_zero`
variable [IsDomain R₁]

theorem separatingLeft_toLinearMap₂'_iff_det_ne_zero {M : Matrix ι ι R₁} :
    M.toLinearMap₂'.SeparatingLeft ↔ M.det ≠ 0 := by
  rw [Matrix.separatingLeft_toLinearMap₂'_iff, Matrix.nondegenerate_iff_det_ne_zero]
#align linear_map.separating_left_to_linear_map₂'_iff_det_ne_zero LinearMap.separatingLeft_toLinearMap₂'_iff_det_ne_zero

theorem separatingLeft_toLinearMap₂'_of_det_ne_zero' (M : Matrix ι ι R₁) (h : M.det ≠ 0) :
    M.toLinearMap₂'.SeparatingLeft :=
  separatingLeft_toLinearMap₂'_iff_det_ne_zero.mpr h
#align linear_map.separating_left_to_linear_map₂'_of_det_ne_zero' LinearMap.separatingLeft_toLinearMap₂'_of_det_ne_zero'

theorem separatingLeft_iff_det_ne_zero {B : M₁ →ₗ[R₁] M₁ →ₗ[R₁] R₁} (b : Basis ι R₁ M₁) :
    B.SeparatingLeft ↔ (toMatrix₂ b b B).det ≠ 0 := by
  rw [← Matrix.nondegenerate_iff_det_ne_zero, nondegenerate_toMatrix_iff]
#align linear_map.separating_left_iff_det_ne_zero LinearMap.separatingLeft_iff_det_ne_zero

theorem separatingLeft_of_det_ne_zero {B : M₁ →ₗ[R₁] M₁ →ₗ[R₁] R₁} (b : Basis ι R₁ M₁)
    (h : (toMatrix₂ b b B).det ≠ 0) : B.SeparatingLeft :=
  (separatingLeft_iff_det_ne_zero b).mpr h
#align linear_map.separating_left_of_det_ne_zero LinearMap.separatingLeft_of_det_ne_zero

end Det

end LinearMap