# Documentation

Mathlib.LinearAlgebra.Matrix.BilinearForm

# Bilinear form #

This file defines the conversion between bilinear forms and matrices.

## Main definitions #

• Matrix.toBilin given a basis define a bilinear form
• Matrix.toBilin' define the bilinear form on n → R
• BilinForm.toMatrix: calculate the matrix coefficients of a bilinear form
• BilinForm.toMatrix': calculate the matrix coefficients of a bilinear form on n → R

## Notations #

In this file we use the following type variables:

• M, M', ... are modules over the semiring R,
• M₁, M₁', ... are modules over the ring R₁,
• M₂, M₂', ... are modules over the commutative semiring R₂,
• M₃, M₃', ... are modules over the commutative ring R₃,
• V, ... is a vector space over the field K.

## Tags #

bilinear form, bilin form, BilinearForm, matrix, basis

def Matrix.toBilin'Aux {R₂ : Type u_5} [] {n : Type u_11} [] (M : Matrix n n R₂) :
BilinForm R₂ (nR₂)

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

This is an auxiliary definition for the equivalence Matrix.toBilin'.

Instances For
theorem Matrix.toBilin'Aux_stdBasis {R₂ : Type u_5} [] {n : Type u_11} [] [] (M : Matrix n n R₂) (i : n) (j : n) :
BilinForm.bilin () (↑(LinearMap.stdBasis R₂ (fun x => R₂) i) 1) (↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1) = M i j
def BilinForm.toMatrixAux {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} (b : nM₂) :
BilinForm R₂ M₂ →ₗ[R₂] Matrix n n R₂

The linear map from bilinear forms to Matrix n n R given an n-indexed basis.

This is an auxiliary definition for the equivalence Matrix.toBilin'.

Instances For
@[simp]
theorem BilinForm.toMatrixAux_apply {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} (B : BilinForm R₂ M₂) (b : nM₂) (i : n) (j : n) :
↑() B i j = BilinForm.bilin B (b i) (b j)
theorem toBilin'Aux_toMatrixAux {R₂ : Type u_5} [] {n : Type u_11} [] [] (B₂ : BilinForm R₂ (nR₂)) :
Matrix.toBilin'Aux (↑(BilinForm.toMatrixAux fun j => ↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1) B₂) = B₂

### ToMatrix' section #

This section deals with the conversion between matrices and bilinear forms on n → R₂.

def BilinForm.toMatrix' {R₂ : Type u_5} [] {n : Type u_11} [] [] :
BilinForm R₂ (nR₂) ≃ₗ[R₂] Matrix n n R₂

The linear equivalence between bilinear forms on n → R and n × n matrices

Instances For
@[simp]
theorem BilinForm.toMatrixAux_stdBasis {R₂ : Type u_5} [] {n : Type u_11} [] [] (B : BilinForm R₂ (nR₂)) :
↑(BilinForm.toMatrixAux fun j => ↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1) B = BilinForm.toMatrix' B
def Matrix.toBilin' {R₂ : Type u_5} [] {n : Type u_11} [] [] :
Matrix n n R₂ ≃ₗ[R₂] BilinForm R₂ (nR₂)

The linear equivalence between n × n matrices and bilinear forms on n → R

Instances For
@[simp]
theorem Matrix.toBilin'Aux_eq {R₂ : Type u_5} [] {n : Type u_11} [] [] (M : Matrix n n R₂) :
= Matrix.toBilin' M
theorem Matrix.toBilin'_apply {R₂ : Type u_5} [] {n : Type u_11} [] [] (M : Matrix n n R₂) (x : nR₂) (y : nR₂) :
BilinForm.bilin (Matrix.toBilin' M) x y = Finset.sum Finset.univ fun i => Finset.sum Finset.univ fun j => x i * M i j * y j
theorem Matrix.toBilin'_apply' {R₂ : Type u_5} [] {n : Type u_11} [] [] (M : Matrix n n R₂) (v : nR₂) (w : nR₂) :
BilinForm.bilin (Matrix.toBilin' M) v w =
@[simp]
theorem Matrix.toBilin'_stdBasis {R₂ : Type u_5} [] {n : Type u_11} [] [] (M : Matrix n n R₂) (i : n) (j : n) :
BilinForm.bilin (Matrix.toBilin' M) (↑(LinearMap.stdBasis R₂ (fun x => R₂) i) 1) (↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1) = M i j
@[simp]
theorem BilinForm.toMatrix'_symm {R₂ : Type u_5} [] {n : Type u_11} [] [] :
LinearEquiv.symm BilinForm.toMatrix' = Matrix.toBilin'
@[simp]
theorem Matrix.toBilin'_symm {R₂ : Type u_5} [] {n : Type u_11} [] [] :
LinearEquiv.symm Matrix.toBilin' = BilinForm.toMatrix'
@[simp]
theorem Matrix.toBilin'_toMatrix' {R₂ : Type u_5} [] {n : Type u_11} [] [] (B : BilinForm R₂ (nR₂)) :
Matrix.toBilin' (BilinForm.toMatrix' B) = B
@[simp]
theorem BilinForm.toMatrix'_toBilin' {R₂ : Type u_5} [] {n : Type u_11} [] [] (M : Matrix n n R₂) :
BilinForm.toMatrix' (Matrix.toBilin' M) = M
@[simp]
theorem BilinForm.toMatrix'_apply {R₂ : Type u_5} [] {n : Type u_11} [] [] (B : BilinForm R₂ (nR₂)) (i : n) (j : n) :
BilinForm.toMatrix' B i j = BilinForm.bilin B (↑(LinearMap.stdBasis R₂ (fun x => R₂) i) 1) (↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1)
@[simp]
theorem BilinForm.toMatrix'_comp {R₂ : Type u_5} [] {n : Type u_11} {o : Type u_12} [] [] [] [] (B : BilinForm R₂ (nR₂)) (l : (oR₂) →ₗ[R₂] nR₂) (r : (oR₂) →ₗ[R₂] nR₂) :
BilinForm.toMatrix' () = Matrix.transpose (LinearMap.toMatrix' l) * BilinForm.toMatrix' B * LinearMap.toMatrix' r
theorem BilinForm.toMatrix'_compLeft {R₂ : Type u_5} [] {n : Type u_11} [] [] (B : BilinForm R₂ (nR₂)) (f : (nR₂) →ₗ[R₂] nR₂) :
BilinForm.toMatrix' () = Matrix.transpose (LinearMap.toMatrix' f) * BilinForm.toMatrix' B
theorem BilinForm.toMatrix'_compRight {R₂ : Type u_5} [] {n : Type u_11} [] [] (B : BilinForm R₂ (nR₂)) (f : (nR₂) →ₗ[R₂] nR₂) :
BilinForm.toMatrix' () = BilinForm.toMatrix' B * LinearMap.toMatrix' f
theorem BilinForm.mul_toMatrix'_mul {R₂ : Type u_5} [] {n : Type u_11} {o : Type u_12} [] [] [] [] (B : BilinForm R₂ (nR₂)) (M : Matrix o n R₂) (N : Matrix n o R₂) :
M * BilinForm.toMatrix' B * N = BilinForm.toMatrix' (BilinForm.comp B (Matrix.toLin' ()) (Matrix.toLin' N))
theorem BilinForm.mul_toMatrix' {R₂ : Type u_5} [] {n : Type u_11} [] [] (B : BilinForm R₂ (nR₂)) (M : Matrix n n R₂) :
M * BilinForm.toMatrix' B = BilinForm.toMatrix' (BilinForm.compLeft B (Matrix.toLin' ()))
theorem BilinForm.toMatrix'_mul {R₂ : Type u_5} [] {n : Type u_11} [] [] (B : BilinForm R₂ (nR₂)) (M : Matrix n n R₂) :
BilinForm.toMatrix' B * M = BilinForm.toMatrix' (BilinForm.compRight B (Matrix.toLin' M))
theorem Matrix.toBilin'_comp {R₂ : Type u_5} [] {n : Type u_11} {o : Type u_12} [] [] [] [] (M : Matrix n n R₂) (P : Matrix n o R₂) (Q : Matrix n o R₂) :
BilinForm.comp (Matrix.toBilin' M) (Matrix.toLin' P) (Matrix.toLin' Q) = Matrix.toBilin' ( * Q)

### ToMatrix section #

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

noncomputable def BilinForm.toMatrix {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) :
BilinForm R₂ M₂ ≃ₗ[R₂] Matrix n n R₂

BilinForm.toMatrix b is the equivalence between R-bilinear forms on M and n-by-n matrices with entries in R, if b is an R-basis for M.

Instances For
noncomputable def Matrix.toBilin {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) :
Matrix n n R₂ ≃ₗ[R₂] BilinForm R₂ M₂

BilinForm.toMatrix b is the equivalence between R-bilinear forms on M and n-by-n matrices with entries in R, if b is an R-basis for M.

Instances For
@[simp]
theorem BilinForm.toMatrix_apply {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) (i : n) (j : n) :
↑() B i j = BilinForm.bilin B (b i) (b j)
@[simp]
theorem Matrix.toBilin_apply {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) (M : Matrix n n R₂) (x : M₂) (y : M₂) :
BilinForm.bilin (↑() M) x y = Finset.sum Finset.univ fun i => Finset.sum Finset.univ fun j => ↑(b.repr x) i * M i j * ↑(b.repr y) j
theorem BilinearForm.toMatrixAux_eq {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) :
↑() B = ↑() B
@[simp]
theorem BilinForm.toMatrix_symm {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) :
@[simp]
theorem Matrix.toBilin_symm {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) :
theorem Matrix.toBilin_basisFun {R₂ : Type u_5} [] {n : Type u_11} [] [] :
Matrix.toBilin (Pi.basisFun R₂ n) = Matrix.toBilin'
theorem BilinForm.toMatrix_basisFun {R₂ : Type u_5} [] {n : Type u_11} [] [] :
= BilinForm.toMatrix'
@[simp]
theorem Matrix.toBilin_toMatrix {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) :
↑() (↑() B) = B
@[simp]
theorem BilinForm.toMatrix_toBilin {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) (M : Matrix n n R₂) :
↑() (↑() M) = M
theorem BilinForm.toMatrix_comp {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} {o : Type u_12} [] [] [] (b : Basis n R₂ M₂) {M₂' : Type u_13} [] [Module R₂ M₂'] (c : Basis o R₂ M₂') [] (B : BilinForm R₂ M₂) (l : M₂' →ₗ[R₂] M₂) (r : M₂' →ₗ[R₂] M₂) :
↑() () = Matrix.transpose (↑() l) * ↑() B * ↑() r
theorem BilinForm.toMatrix_compLeft {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) (f : M₂ →ₗ[R₂] M₂) :
↑() () = Matrix.transpose (↑() f) * ↑() B
theorem BilinForm.toMatrix_compRight {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) (f : M₂ →ₗ[R₂] M₂) :
↑() () = ↑() B * ↑() f
@[simp]
theorem BilinForm.toMatrix_mul_basis_toMatrix {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} {o : Type u_12} [] [] [] (b : Basis n R₂ M₂) [] (c : Basis o R₂ M₂) (B : BilinForm R₂ M₂) :
* ↑() B * = ↑() B
theorem BilinForm.mul_toMatrix_mul {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} {o : Type u_12} [] [] [] (b : Basis n R₂ M₂) {M₂' : Type u_13} [] [Module R₂ M₂'] (c : Basis o R₂ M₂') [] (B : BilinForm R₂ M₂) (M : Matrix o n R₂) (N : Matrix n o R₂) :
M * ↑() B * N = ↑() (BilinForm.comp B (↑() ()) (↑() N))
theorem BilinForm.mul_toMatrix {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) (M : Matrix n n R₂) :
M * ↑() B = ↑() (BilinForm.compLeft B (↑() ()))
theorem BilinForm.toMatrix_mul {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} [] [] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) (M : Matrix n n R₂) :
↑() B * M = ↑() (BilinForm.compRight B (↑() M))
theorem Matrix.toBilin_comp {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {n : Type u_11} {o : Type u_12} [] [] [] (b : Basis n R₂ M₂) {M₂' : Type u_13} [] [Module R₂ M₂'] (c : Basis o R₂ M₂') [] (M : Matrix n n R₂) (P : Matrix n o R₂) (Q : Matrix n o R₂) :
BilinForm.comp (↑() M) (↑() P) (↑() Q) = ↑() ( * Q)
@[simp]
theorem isAdjointPair_toBilin' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [] (J : Matrix n n R₃) (J₃ : Matrix n n R₃) (A : Matrix n n R₃) (A' : Matrix n n R₃) [] :
BilinForm.IsAdjointPair (Matrix.toBilin' J) (Matrix.toBilin' J₃) (Matrix.toLin' A) (Matrix.toLin' A') Matrix.IsAdjointPair J J₃ A A'
@[simp]
theorem isAdjointPair_toBilin {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [] [Module R₃ M₃] {n : Type u_11} [] (b : Basis n R₃ M₃) (J : Matrix n n R₃) (J₃ : Matrix n n R₃) (A : Matrix n n R₃) (A' : Matrix n n R₃) [] :
BilinForm.IsAdjointPair (↑() J) (↑() J₃) (↑() A) (↑() A') Matrix.IsAdjointPair J J₃ A A'
theorem Matrix.isAdjointPair_equiv' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [] (J : Matrix n n R₃) (A : Matrix n n R₃) (A' : Matrix n n R₃) [] (P : Matrix n n R₃) (h : ) :
Matrix.IsAdjointPair ( * P) ( * P) A A' Matrix.IsAdjointPair J J (P * A * P⁻¹) (P * A' * P⁻¹)
def pairSelfAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [] (J : Matrix n n R₃) (J₃ : Matrix n n R₃) [] :
Submodule R₃ (Matrix n n R₃)

The submodule of pair-self-adjoint matrices with respect to bilinear forms corresponding to given matrices J, J₂.

Instances For
theorem mem_pairSelfAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [] (J : Matrix n n R₃) (J₃ : Matrix n n R₃) (A : Matrix n n R₃) [] :
def selfAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [] (J : Matrix n n R₃) [] :
Submodule R₃ (Matrix n n R₃)

The submodule of self-adjoint matrices with respect to the bilinear form corresponding to the matrix J.

Instances For
theorem mem_selfAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [] (J : Matrix n n R₃) (A : Matrix n n R₃) [] :
def skewAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [] (J : Matrix n n R₃) [] :
Submodule R₃ (Matrix n n R₃)

The submodule of skew-adjoint matrices with respect to the bilinear form corresponding to the matrix J.

Instances For
theorem mem_skewAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [] (J : Matrix n n R₃) (A : Matrix n n R₃) [] :
theorem Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin {R₂ : Type u_5} {M₂ : Type u_6} [] [] [Module R₂ M₂] {ι : Type u_12} [] [] {M : Matrix ι ι R₂} (b : Basis ι R₂ M₂) :
BilinForm.Nondegenerate (Matrix.toBilin' M)
theorem Matrix.Nondegenerate.toBilin' {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [] [] {M : Matrix ι ι R₃} (h : ) :
BilinForm.Nondegenerate (Matrix.toBilin' M)
@[simp]
theorem Matrix.nondegenerate_toBilin'_iff {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [] [] {M : Matrix ι ι R₃} :
BilinForm.Nondegenerate (Matrix.toBilin' M)
theorem Matrix.Nondegenerate.toBilin {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [] [Module R₃ M₃] {ι : Type u_12} [] [] {M : Matrix ι ι R₃} (h : ) (b : Basis ι R₃ M₃) :
@[simp]
theorem Matrix.nondegenerate_toBilin_iff {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [] [Module R₃ M₃] {ι : Type u_12} [] [] {M : Matrix ι ι R₃} (b : Basis ι R₃ M₃) :
@[simp]
theorem BilinForm.nondegenerate_toMatrix'_iff {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [] [] {B : BilinForm R₃ (ιR₃)} :
Matrix.Nondegenerate (BilinForm.toMatrix' B)
theorem BilinForm.Nondegenerate.toMatrix' {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [] [] {B : BilinForm R₃ (ιR₃)} (h : ) :
Matrix.Nondegenerate (BilinForm.toMatrix' B)
@[simp]
theorem BilinForm.nondegenerate_toMatrix_iff {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [] [Module R₃ M₃] {ι : Type u_12} [] [] {B : BilinForm R₃ M₃} (b : Basis ι R₃ M₃) :
theorem BilinForm.Nondegenerate.toMatrix {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [] [Module R₃ M₃] {ι : Type u_12} [] [] {B : BilinForm R₃ M₃} (h : ) (b : Basis ι R₃ M₃) :
theorem BilinForm.nondegenerate_toBilin'_iff_det_ne_zero {A : Type u_11} [] [] {ι : Type u_12} [] [] {M : Matrix ι ι A} :
BilinForm.Nondegenerate (Matrix.toBilin' M) 0
theorem BilinForm.nondegenerate_toBilin'_of_det_ne_zero' {A : Type u_11} [] [] {ι : Type u_12} [] [] (M : Matrix ι ι A) (h : 0) :
BilinForm.Nondegenerate (Matrix.toBilin' M)
theorem BilinForm.nondegenerate_iff_det_ne_zero {M₃ : Type u_8} [] {A : Type u_11} [] [] [Module A M₃] {ι : Type u_12} [] [] {B : BilinForm A M₃} (b : Basis ι A M₃) :
Matrix.det (↑() B) 0
theorem BilinForm.nondegenerate_of_det_ne_zero {M₃ : Type u_8} [] {A : Type u_11} [] [] [Module A M₃] (B₃ : BilinForm A M₃) {ι : Type u_12} [] [] (b : Basis ι A M₃) (h : Matrix.det (↑() B₃) 0) :