# Bilinear form #

This file defines various properties of bilinear forms, including reflexivity, symmetry, alternativity, adjoint, and non-degeneracy. For orthogonality, see LinearAlgebra/BilinearForm/Orthogonal.lean.

## Notations #

Given any term B of type BilinForm, due to a coercion, can use the notation B x y to refer to the function field, ie. B x y = B.bilin x y.

In this file we use the following type variables:

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

## Tags #

Bilinear form,

### Reflexivity, symmetry, and alternativity #

def LinearMap.BilinForm.IsRefl {R : Type u_1} {M : Type u_2} [] [] [Module R M] (B : ) :

The proposition that a bilinear form is reflexive

Equations
• = ∀ (x y : M), (B x) y = 0(B y) x = 0
Instances For
theorem LinearMap.BilinForm.IsRefl.eq_zero {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } (H : ) {x : M} {y : M} :
(B x) y = 0(B y) x = 0
theorem LinearMap.BilinForm.IsRefl.neg {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] {B : } (hB : ) :
theorem LinearMap.BilinForm.IsRefl.smul {R : Type u_1} {M : Type u_2} [] [] [Module R M] {α : Type u_9} [] [Module α R] [] [] (a : α) {B : } (hB : ) :
theorem LinearMap.BilinForm.IsRefl.groupSMul {R : Type u_1} {M : Type u_2} [] [] [Module R M] {α : Type u_9} [] [] [] (a : α) {B : } (hB : ) :
@[simp]
theorem LinearMap.BilinForm.isRefl_zero {R : Type u_1} {M : Type u_2} [] [] [Module R M] :
@[simp]
theorem LinearMap.BilinForm.isRefl_neg {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] {B : } :
def LinearMap.BilinForm.IsSymm {R : Type u_1} {M : Type u_2} [] [] [Module R M] (B : ) :

The proposition that a bilinear form is symmetric

Equations
• = ∀ (x y : M), (B x) y = (B y) x
Instances For
theorem LinearMap.BilinForm.IsSymm.eq {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } (H : ) (x : M) (y : M) :
(B x) y = (B y) x
theorem LinearMap.BilinForm.IsSymm.isRefl {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } (H : ) :
theorem LinearMap.BilinForm.IsSymm.add {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B₁ : } {B₂ : } (hB₁ : ) (hB₂ : ) :
theorem LinearMap.BilinForm.IsSymm.sub {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] {B₁ : } {B₂ : } (hB₁ : ) (hB₂ : ) :
theorem LinearMap.BilinForm.IsSymm.neg {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] {B : } (hB : ) :
theorem LinearMap.BilinForm.IsSymm.smul {R : Type u_1} {M : Type u_2} [] [] [Module R M] {α : Type u_9} [] [] [] (a : α) {B : } (hB : ) :
theorem LinearMap.BilinForm.IsSymm.restrict {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } (b : ) (W : ) :

The restriction of a symmetric bilinear form on a submodule is also symmetric.

@[simp]
theorem LinearMap.BilinForm.isSymm_zero {R : Type u_1} {M : Type u_2} [] [] [Module R M] :
@[simp]
theorem LinearMap.BilinForm.isSymm_neg {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] {B : } :
theorem LinearMap.BilinForm.isSymm_iff_flip {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } :
LinearMap.BilinForm.flipHom B = B
def LinearMap.BilinForm.IsAlt {R : Type u_1} {M : Type u_2} [] [] [Module R M] (B : ) :

The proposition that a bilinear form is alternating

Equations
• = ∀ (x : M), (B x) x = 0
Instances For
theorem LinearMap.BilinForm.IsAlt.self_eq_zero {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } (H : ) (x : M) :
(B x) x = 0
theorem LinearMap.BilinForm.IsAlt.neg_eq {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] {B₁ : } (H : ) (x : M₁) (y : M₁) :
-(B₁ x) y = (B₁ y) x
theorem LinearMap.BilinForm.IsAlt.isRefl {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] {B₁ : } (H : ) :
theorem LinearMap.BilinForm.IsAlt.add {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B₁ : } {B₂ : } (hB₁ : ) (hB₂ : ) :
theorem LinearMap.BilinForm.IsAlt.sub {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] {B₁ : } {B₂ : } (hB₁ : ) (hB₂ : ) :
theorem LinearMap.BilinForm.IsAlt.neg {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] {B : } (hB : ) :
theorem LinearMap.BilinForm.IsAlt.smul {R : Type u_1} {M : Type u_2} [] [] [Module R M] {α : Type u_9} [] [] [] (a : α) {B : } (hB : ) :
@[simp]
theorem LinearMap.BilinForm.isAlt_zero {R : Type u_1} {M : Type u_2} [] [] [Module R M] :
@[simp]
theorem LinearMap.BilinForm.isAlt_neg {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] {B : } :

def LinearMap.BilinForm.IsAdjointPair {R : Type u_1} {M : Type u_2} [] [] [Module R M] (B : ) {M' : Type u_9} [] [Module R M'] (B' : ) (f : M →ₗ[R] M') (g : M' →ₗ[R] M) :

Given a pair of modules equipped with bilinear forms, this is the condition for a pair of maps between them to be mutually adjoint.

Equations
• = ∀ ⦃x : M⦄ ⦃y : M'⦄, (B' (f x)) y = (B x) (g y)
Instances For
theorem LinearMap.BilinForm.IsAdjointPair.eq {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } {M' : Type u_9} [] [Module R M'] {B' : } {f : M →ₗ[R] M'} {g : M' →ₗ[R] M} (h : ) {x : M} {y : M'} :
(B' (f x)) y = (B x) (g y)
theorem LinearMap.BilinForm.isAdjointPair_iff_compLeft_eq_compRight {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } (F : ) (f : ) (g : ) :
theorem LinearMap.BilinForm.isAdjointPair_zero {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } {M' : Type u_9} [] [Module R M'] {B' : } :
theorem LinearMap.BilinForm.isAdjointPair_id {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } :
theorem LinearMap.BilinForm.IsAdjointPair.add {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } {M' : Type u_9} [] [Module R M'] {B' : } {f : M →ₗ[R] M'} {f' : M →ₗ[R] M'} {g : M' →ₗ[R] M} {g' : M' →ₗ[R] M} (h : ) (h' : ) :
theorem LinearMap.BilinForm.IsAdjointPair.sub {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] {B₁ : } {M₁' : Type u_10} [AddCommGroup M₁'] [Module R₁ M₁'] {B₁' : LinearMap.BilinForm R₁ M₁'} {f₁ : M₁ →ₗ[R₁] M₁'} {f₁' : M₁ →ₗ[R₁] M₁'} {g₁ : M₁' →ₗ[R₁] M₁} {g₁' : M₁' →ₗ[R₁] M₁} (h : LinearMap.BilinForm.IsAdjointPair B₁ B₁' f₁ g₁) (h' : LinearMap.BilinForm.IsAdjointPair B₁ B₁' f₁' g₁') :
LinearMap.BilinForm.IsAdjointPair B₁ B₁' (f₁ - f₁') (g₁ - g₁')
theorem LinearMap.BilinForm.IsAdjointPair.smul {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } {M' : Type u_9} [] [Module R M'] {B₂' : } {f₂ : M →ₗ[R] M'} {g₂ : M' →ₗ[R] M} (c : R) (h : ) :
LinearMap.BilinForm.IsAdjointPair B B₂' (c f₂) (c g₂)
theorem LinearMap.BilinForm.IsAdjointPair.comp {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } {M' : Type u_9} [] [Module R M'] {B' : } {f : M →ₗ[R] M'} {g : M' →ₗ[R] M} {M'' : Type u_11} [] [Module R M''] (B'' : ) {f' : M' →ₗ[R] M''} {g' : M'' →ₗ[R] M'} (h : ) (h' : LinearMap.BilinForm.IsAdjointPair B' B'' f' g') :
LinearMap.BilinForm.IsAdjointPair B B'' (f' ∘ₗ f) (g ∘ₗ g')
theorem LinearMap.BilinForm.IsAdjointPair.mul {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } {f : } {g : } {f' : } {g' : } (h : ) (h' : ) :
def LinearMap.BilinForm.IsPairSelfAdjoint {R : Type u_1} {M : Type u_2} [] [] [Module R M] (B : ) (F : ) (f : ) :

The condition for an endomorphism to be "self-adjoint" with respect to a pair of bilinear forms on the underlying module. In the case that these two forms are identical, this is the usual concept of self adjointness. In the case that one of the forms is the negation of the other, this is the usual concept of skew adjointness.

Equations
Instances For
def LinearMap.BilinForm.isPairSelfAdjointSubmodule {R : Type u_1} {M : Type u_2} [] [] [Module R M] (B₂ : ) (F₂ : ) :

The set of pair-self-adjoint endomorphisms are a submodule of the type of all endomorphisms.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem LinearMap.BilinForm.mem_isPairSelfAdjointSubmodule {R : Type u_1} {M : Type u_2} [] [] [Module R M] (B₂ : ) (F₂ : ) (f : ) :
theorem LinearMap.BilinForm.isPairSelfAdjoint_equiv {R : Type u_1} {M : Type u_2} [] [] [Module R M] {M' : Type u_9} [] [Module R M'] (B₂ : ) (F₂ : ) (e : M' ≃ₗ[R] M) (f : ) :
def LinearMap.BilinForm.IsSelfAdjoint {R : Type u_1} {M : Type u_2} [] [] [Module R M] (B : ) (f : ) :

An endomorphism of a module is self-adjoint with respect to a bilinear form if it serves as an adjoint for itself.

Equations
Instances For
def LinearMap.BilinForm.IsSkewAdjoint {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] (B₁ : ) (f : Module.End R₁ M₁) :

An endomorphism of a module is skew-adjoint with respect to a bilinear form if its negation serves as an adjoint.

Equations
Instances For
theorem LinearMap.BilinForm.isSkewAdjoint_iff_neg_self_adjoint {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] (B₁ : ) (f : Module.End R₁ M₁) :
def LinearMap.BilinForm.selfAdjointSubmodule {R : Type u_1} {M : Type u_2} [] [] [Module R M] (B : ) :

The set of self-adjoint endomorphisms of a module with bilinear form is a submodule. (In fact it is a Jordan subalgebra.)

Equations
Instances For
@[simp]
theorem LinearMap.BilinForm.mem_selfAdjointSubmodule {R : Type u_1} {M : Type u_2} [] [] [Module R M] (B : ) (f : ) :
def LinearMap.BilinForm.skewAdjointSubmodule {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] (B₁ : ) :
Submodule R₁ (Module.End R₁ M₁)

The set of skew-adjoint endomorphisms of a module with bilinear form is a submodule. (In fact it is a Lie subalgebra.)

Equations
Instances For
@[simp]
theorem LinearMap.BilinForm.mem_skewAdjointSubmodule {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] (B₁ : ) (f : Module.End R₁ M₁) :
def LinearMap.BilinForm.Nondegenerate {R : Type u_1} {M : Type u_2} [] [] [Module R M] (B : ) :

A nondegenerate bilinear form is a bilinear form such that the only element that is orthogonal to every other element is 0; i.e., for all nonzero m in M, there exists n in M with B m n ≠ 0.

Note that for general (neither symmetric nor antisymmetric) bilinear forms this definition has a chirality; in addition to this "left" nondegeneracy condition one could define a "right" nondegeneracy condition that in the situation described, B n m ≠ 0. This variant definition is not currently provided in mathlib. In finite dimension either definition implies the other.

Equations
• = ∀ (m : M), (∀ (n : M), (B m) n = 0)m = 0
Instances For
theorem LinearMap.BilinForm.not_nondegenerate_zero (R : Type u_1) (M : Type u_2) [] [] [Module R M] [] :

In a non-trivial module, zero is not non-degenerate.

theorem LinearMap.BilinForm.Nondegenerate.ne_zero {R : Type u_1} {M : Type u_2} [] [] [Module R M] [] {B : } :
B 0
theorem LinearMap.BilinForm.Nondegenerate.congr {R : Type u_1} {M : Type u_2} [] [] [Module R M] {M' : Type u_9} [] [Module R M'] {B : } (e : M ≃ₗ[R] M') :
@[simp]
theorem LinearMap.BilinForm.nondegenerate_congr_iff {R : Type u_1} {M : Type u_2} [] [] [Module R M] {M' : Type u_9} [] [Module R M'] {B : } (e : M ≃ₗ[R] M') :
theorem LinearMap.BilinForm.nondegenerate_iff_ker_eq_bot {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } :

A bilinear form is nondegenerate if and only if it has a trivial kernel.

theorem LinearMap.BilinForm.Nondegenerate.ker_eq_bot {R : Type u_1} {M : Type u_2} [] [] [Module R M] {B : } :
LinearMap.ker (LinearMap.BilinForm.toLin B) =
theorem LinearMap.BilinForm.compLeft_injective {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] (B : ) :
theorem LinearMap.BilinForm.isAdjointPair_unique_of_nondegenerate {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [] [Module R₁ M₁] (B : ) (φ : M₁ →ₗ[R₁] M₁) (ψ₁ : M₁ →ₗ[R₁] M₁) (ψ₂ : M₁ →ₗ[R₁] M₁) (hψ₁ : ) (hψ₂ : ) :
ψ₁ = ψ₂
noncomputable def LinearMap.BilinForm.toDual {V : Type u_5} {K : Type u_6} [] [] [Module K V] [] (B : ) :

Given a nondegenerate bilinear form B on a finite-dimensional vector space, B.toDual is the linear equivalence between a vector space and its dual with the underlying linear map B.toLin.

Equations
Instances For
theorem LinearMap.BilinForm.toDual_def {V : Type u_5} {K : Type u_6} [] [] [Module K V] [] {B : } (b : ) {m : V} {n : V} :
( m) n = (B m) n
theorem LinearMap.BilinForm.Nondegenerate.flip {V : Type u_5} {K : Type u_6} [] [] [Module K V] [] {B : } :
theorem LinearMap.BilinForm.nonDegenerateFlip_iff {V : Type u_5} {K : Type u_6} [] [] [Module K V] [] {B : } :
noncomputable def LinearMap.BilinForm.dualBasis {V : Type u_5} {K : Type u_6} [] [] [Module K V] {ι : Type u_10} [] [] (B : ) (b : Basis ι K V) :
Basis ι K V

The B-dual basis B.dualBasis hB b to a finite basis b satisfies B (B.dualBasis hB b i) (b j) = B (b i) (B.dualBasis hB b j) = if i = j then 1 else 0, where B is a nondegenerate (symmetric) bilinear form and b is a finite basis.

Equations
Instances For
@[simp]
theorem LinearMap.BilinForm.dualBasis_repr_apply {V : Type u_5} {K : Type u_6} [] [] [Module K V] {ι : Type u_10} [] [] (B : ) (b : Basis ι K V) (x : V) (i : ι) :
(().repr x) i = (B x) (b i)
theorem LinearMap.BilinForm.apply_dualBasis_left {V : Type u_5} {K : Type u_6} [] [] [Module K V] {ι : Type u_10} [] [] (B : ) (b : Basis ι K V) (i : ι) (j : ι) :
(B (() i)) (b j) = if j = i then 1 else 0
theorem LinearMap.BilinForm.apply_dualBasis_right {V : Type u_5} {K : Type u_6} [] [] [Module K V] {ι : Type u_10} [] [] (B : ) (sym : ) (b : Basis ι K V) (i : ι) (j : ι) :
(B (b i)) (() j) = if i = j then 1 else 0
@[simp]
theorem LinearMap.BilinForm.dualBasis_dualBasis_flip {V : Type u_5} {K : Type u_6} [] [] [Module K V] [] (B : ) {ι : Type u_11} [] [] (b : Basis ι K V) :
@[simp]
theorem LinearMap.BilinForm.dualBasis_flip_dualBasis {V : Type u_5} {K : Type u_6} [] [] [Module K V] (B : ) {ι : Type u_11} [] [] [] (b : Basis ι K V) :
@[simp]
theorem LinearMap.BilinForm.dualBasis_dualBasis {V : Type u_5} {K : Type u_6} [] [] [Module K V] (B : ) (hB' : ) {ι : Type u_11} [] [] [] (b : Basis ι K V) :
= b
noncomputable def LinearMap.BilinForm.symmCompOfNondegenerate {V : Type u_5} {K : Type u_6} [] [] [Module K V] [] (B₁ : ) (B₂ : ) (b₂ : ) :

Given bilinear forms B₁, B₂ where B₂ is nondegenerate, symmCompOfNondegenerate is the linear map B₂.toLin⁻¹ ∘ B₁.toLin.

Equations
Instances For
theorem LinearMap.BilinForm.comp_symmCompOfNondegenerate_apply {V : Type u_5} {K : Type u_6} [] [] [Module K V] [] (B₁ : ) {B₂ : } (b₂ : ) (v : V) :
B₂ (() v) = B₁ v
@[simp]
theorem LinearMap.BilinForm.symmCompOfNondegenerate_left_apply {V : Type u_5} {K : Type u_6} [] [] [Module K V] [] (B₁ : ) {B₂ : } (b₂ : ) (v : V) (w : V) :
(B₂ (() w)) v = (B₁ w) v
noncomputable def LinearMap.BilinForm.leftAdjointOfNondegenerate {V : Type u_5} {K : Type u_6} [] [] [Module K V] [] (B : ) (φ : V →ₗ[K] V) :

Given the nondegenerate bilinear form B and the linear map φ, leftAdjointOfNondegenerate provides the left adjoint of φ with respect to B. The lemma proving this property is BilinForm.isAdjointPairLeftAdjointOfNondegenerate.

Equations
Instances For
theorem LinearMap.BilinForm.isAdjointPairLeftAdjointOfNondegenerate {V : Type u_5} {K : Type u_6} [] [] [Module K V] [] (B : ) (φ : V →ₗ[K] V) :
theorem LinearMap.BilinForm.isAdjointPair_iff_eq_of_nondegenerate {V : Type u_5} {K : Type u_6} [] [] [Module K V] [] (B : ) (ψ : V →ₗ[K] V) (φ : V →ₗ[K] V) :

Given the nondegenerate bilinear form B, the linear map φ has a unique left adjoint given by BilinForm.leftAdjointOfNondegenerate.