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 semiringR,M₁,M₁', ... are modules over the commutative ringR₁,V, ... is a vector space over the fieldK.
References #
Tags #
Bilinear form,
Reflexivity, symmetry, and alternativity #
def
BilinForm.IsRefl
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
(B : BilinForm R M)
:
The proposition that a bilinear form is reflexive
Equations
- BilinForm.IsRefl B = ∀ (x y : M), B.bilin x y = 0 → B.bilin y x = 0
theorem
BilinForm.IsRefl.eq_zero
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
(H : BilinForm.IsRefl B)
{x : M}
{y : M}
:
theorem
BilinForm.IsRefl.neg
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{B : BilinForm R₁ M₁}
(hB : BilinForm.IsRefl B)
:
theorem
BilinForm.IsRefl.smul
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{α : Type u_9}
[CommSemiring α]
[Module α R]
[SMulCommClass α R R]
[NoZeroSMulDivisors α R]
(a : α)
{B : BilinForm R M}
(hB : BilinForm.IsRefl B)
:
BilinForm.IsRefl (a • B)
theorem
BilinForm.IsRefl.groupSMul
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{α : Type u_9}
[Group α]
[DistribMulAction α R]
[SMulCommClass α R R]
(a : α)
{B : BilinForm R M}
(hB : BilinForm.IsRefl B)
:
BilinForm.IsRefl (a • B)
@[simp]
theorem
BilinForm.isRefl_zero
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
:
@[simp]
theorem
BilinForm.isRefl_neg
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{B : BilinForm R₁ M₁}
:
def
BilinForm.IsSymm
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
(B : BilinForm R M)
:
The proposition that a bilinear form is symmetric
Equations
- BilinForm.IsSymm B = ∀ (x y : M), B.bilin x y = B.bilin y x
theorem
BilinForm.IsSymm.eq
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
(H : BilinForm.IsSymm B)
(x : M)
(y : M)
:
B.bilin x y = B.bilin y x
theorem
BilinForm.IsSymm.isRefl
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
(H : BilinForm.IsSymm B)
:
theorem
BilinForm.IsSymm.add
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B₁ : BilinForm R M}
{B₂ : BilinForm R M}
(hB₁ : BilinForm.IsSymm B₁)
(hB₂ : BilinForm.IsSymm B₂)
:
BilinForm.IsSymm (B₁ + B₂)
theorem
BilinForm.IsSymm.sub
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{B₁ : BilinForm R₁ M₁}
{B₂ : BilinForm R₁ M₁}
(hB₁ : BilinForm.IsSymm B₁)
(hB₂ : BilinForm.IsSymm B₂)
:
BilinForm.IsSymm (B₁ - B₂)
theorem
BilinForm.IsSymm.neg
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{B : BilinForm R₁ M₁}
(hB : BilinForm.IsSymm B)
:
theorem
BilinForm.IsSymm.smul
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{α : Type u_9}
[Monoid α]
[DistribMulAction α R]
[SMulCommClass α R R]
(a : α)
{B : BilinForm R M}
(hB : BilinForm.IsSymm B)
:
BilinForm.IsSymm (a • B)
theorem
BilinForm.IsSymm.restrict
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
(b : BilinForm.IsSymm B)
(W : Submodule R M)
:
The restriction of a symmetric bilinear form on a submodule is also symmetric.
@[simp]
theorem
BilinForm.isSymm_zero
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
:
@[simp]
theorem
BilinForm.isSymm_neg
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{B : BilinForm R₁ M₁}
:
theorem
BilinForm.isSymm_iff_flip
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
:
BilinForm.IsSymm B ↔ BilinForm.flipHom B = B
def
BilinForm.IsAlt
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
(B : BilinForm R M)
:
The proposition that a bilinear form is alternating
Equations
- BilinForm.IsAlt B = ∀ (x : M), B.bilin x x = 0
theorem
BilinForm.IsAlt.self_eq_zero
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
(H : BilinForm.IsAlt B)
(x : M)
:
B.bilin x x = 0
theorem
BilinForm.IsAlt.neg_eq
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{B₁ : BilinForm R₁ M₁}
(H : BilinForm.IsAlt B₁)
(x : M₁)
(y : M₁)
:
theorem
BilinForm.IsAlt.isRefl
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{B₁ : BilinForm R₁ M₁}
(H : BilinForm.IsAlt B₁)
:
theorem
BilinForm.IsAlt.add
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B₁ : BilinForm R M}
{B₂ : BilinForm R M}
(hB₁ : BilinForm.IsAlt B₁)
(hB₂ : BilinForm.IsAlt B₂)
:
BilinForm.IsAlt (B₁ + B₂)
theorem
BilinForm.IsAlt.sub
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{B₁ : BilinForm R₁ M₁}
{B₂ : BilinForm R₁ M₁}
(hB₁ : BilinForm.IsAlt B₁)
(hB₂ : BilinForm.IsAlt B₂)
:
BilinForm.IsAlt (B₁ - B₂)
theorem
BilinForm.IsAlt.neg
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{B : BilinForm R₁ M₁}
(hB : BilinForm.IsAlt B)
:
BilinForm.IsAlt (-B)
theorem
BilinForm.IsAlt.smul
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{α : Type u_9}
[Monoid α]
[DistribMulAction α R]
[SMulCommClass α R R]
(a : α)
{B : BilinForm R M}
(hB : BilinForm.IsAlt B)
:
BilinForm.IsAlt (a • B)
@[simp]
theorem
BilinForm.isAlt_zero
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
:
@[simp]
theorem
BilinForm.isAlt_neg
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{B : BilinForm R₁ M₁}
:
BilinForm.IsAlt (-B) ↔ BilinForm.IsAlt B
Linear adjoints #
def
BilinForm.IsAdjointPair
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
(B : BilinForm R M)
{M' : Type u_9}
[AddCommMonoid M']
[Module R M']
(B' : BilinForm R M')
(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
- BilinForm.IsAdjointPair B B' f g = ∀ ⦃x : M⦄ ⦃y : M'⦄, B'.bilin (f x) y = B.bilin x (g y)
theorem
BilinForm.IsAdjointPair.eq
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
{M' : Type u_9}
[AddCommMonoid M']
[Module R M']
{B' : BilinForm R M'}
{f : M →ₗ[R] M'}
{g : M' →ₗ[R] M}
(h : BilinForm.IsAdjointPair B B' f g)
{x : M}
{y : M'}
:
B'.bilin (f x) y = B.bilin x (g y)
theorem
BilinForm.isAdjointPair_iff_compLeft_eq_compRight
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
(F : BilinForm R M)
(f : Module.End R M)
(g : Module.End R M)
:
BilinForm.IsAdjointPair B F f g ↔ BilinForm.compLeft F f = BilinForm.compRight B g
theorem
BilinForm.isAdjointPair_zero
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
{M' : Type u_9}
[AddCommMonoid M']
[Module R M']
{B' : BilinForm R M'}
:
BilinForm.IsAdjointPair B B' 0 0
theorem
BilinForm.isAdjointPair_id
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
:
BilinForm.IsAdjointPair B B 1 1
theorem
BilinForm.IsAdjointPair.add
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
{M' : Type u_9}
[AddCommMonoid M']
[Module R M']
{B' : BilinForm R M'}
{f : M →ₗ[R] M'}
{f' : M →ₗ[R] M'}
{g : M' →ₗ[R] M}
{g' : M' →ₗ[R] M}
(h : BilinForm.IsAdjointPair B B' f g)
(h' : BilinForm.IsAdjointPair B B' f' g')
:
BilinForm.IsAdjointPair B B' (f + f') (g + g')
theorem
BilinForm.IsAdjointPair.sub
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{B₁ : BilinForm R₁ M₁}
{M₁' : Type u_10}
[AddCommGroup M₁']
[Module R₁ M₁']
{B₁' : BilinForm R₁ M₁'}
{f₁ : M₁ →ₗ[R₁] M₁'}
{f₁' : M₁ →ₗ[R₁] M₁'}
{g₁ : M₁' →ₗ[R₁] M₁}
{g₁' : M₁' →ₗ[R₁] M₁}
(h : BilinForm.IsAdjointPair B₁ B₁' f₁ g₁)
(h' : BilinForm.IsAdjointPair B₁ B₁' f₁' g₁')
:
BilinForm.IsAdjointPair B₁ B₁' (f₁ - f₁') (g₁ - g₁')
theorem
BilinForm.IsAdjointPair.smul
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
{M' : Type u_9}
[AddCommMonoid M']
[Module R M']
{B₂' : BilinForm R M'}
{f₂ : M →ₗ[R] M'}
{g₂ : M' →ₗ[R] M}
(c : R)
(h : BilinForm.IsAdjointPair B B₂' f₂ g₂)
:
BilinForm.IsAdjointPair B B₂' (c • f₂) (c • g₂)
theorem
BilinForm.IsAdjointPair.comp
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
{M' : Type u_9}
[AddCommMonoid M']
[Module R M']
{B' : BilinForm R M'}
{f : M →ₗ[R] M'}
{g : M' →ₗ[R] M}
{M'' : Type u_11}
[AddCommMonoid M'']
[Module R M'']
(B'' : BilinForm R M'')
{f' : M' →ₗ[R] M''}
{g' : M'' →ₗ[R] M'}
(h : BilinForm.IsAdjointPair B B' f g)
(h' : BilinForm.IsAdjointPair B' B'' f' g')
:
BilinForm.IsAdjointPair B B'' (f' ∘ₗ f) (g ∘ₗ g')
theorem
BilinForm.IsAdjointPair.mul
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
{f : Module.End R M}
{g : Module.End R M}
{f' : Module.End R M}
{g' : Module.End R M}
(h : BilinForm.IsAdjointPair B B f g)
(h' : BilinForm.IsAdjointPair B B f' g')
:
BilinForm.IsAdjointPair B B (f * f') (g' * g)
def
BilinForm.IsPairSelfAdjoint
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
(B : BilinForm R M)
(F : BilinForm R M)
(f : Module.End R M)
:
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
- BilinForm.IsPairSelfAdjoint B F f = BilinForm.IsAdjointPair B F f f
def
BilinForm.isPairSelfAdjointSubmodule
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
(B₂ : BilinForm R M)
(F₂ : BilinForm R M)
:
Submodule R (Module.End R M)
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.
@[simp]
theorem
BilinForm.mem_isPairSelfAdjointSubmodule
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
(B₂ : BilinForm R M)
(F₂ : BilinForm R M)
(f : Module.End R M)
:
f ∈ BilinForm.isPairSelfAdjointSubmodule B₂ F₂ ↔ BilinForm.IsPairSelfAdjoint B₂ F₂ f
theorem
BilinForm.isPairSelfAdjoint_equiv
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{M' : Type u_9}
[AddCommMonoid M']
[Module R M']
(B₂ : BilinForm R M)
(F₂ : BilinForm R M)
(e : M' ≃ₗ[R] M)
(f : Module.End R M)
:
BilinForm.IsPairSelfAdjoint B₂ F₂ f ↔ BilinForm.IsPairSelfAdjoint (BilinForm.comp B₂ ↑e ↑e) (BilinForm.comp F₂ ↑e ↑e)
((LinearEquiv.conj (LinearEquiv.symm e)) f)
def
BilinForm.IsSelfAdjoint
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
(B : BilinForm R M)
(f : Module.End R M)
:
An endomorphism of a module is self-adjoint with respect to a bilinear form if it serves as an adjoint for itself.
Equations
- BilinForm.IsSelfAdjoint B f = BilinForm.IsAdjointPair B B f f
def
BilinForm.IsSkewAdjoint
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
(B₁ : BilinForm R₁ M₁)
(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
- BilinForm.IsSkewAdjoint B₁ f = BilinForm.IsAdjointPair B₁ B₁ f (-f)
theorem
BilinForm.isSkewAdjoint_iff_neg_self_adjoint
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
(B₁ : BilinForm R₁ M₁)
(f : Module.End R₁ M₁)
:
BilinForm.IsSkewAdjoint B₁ f ↔ BilinForm.IsAdjointPair (-B₁) B₁ f f
def
BilinForm.selfAdjointSubmodule
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
(B : BilinForm R M)
:
Submodule R (Module.End R M)
The set of self-adjoint endomorphisms of a module with bilinear form is a submodule. (In fact it is a Jordan subalgebra.)
Equations
@[simp]
theorem
BilinForm.mem_selfAdjointSubmodule
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
(B : BilinForm R M)
(f : Module.End R M)
:
def
BilinForm.skewAdjointSubmodule
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
(B₁ : BilinForm R₁ M₁)
:
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
@[simp]
theorem
BilinForm.mem_skewAdjointSubmodule
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
(B₁ : BilinForm R₁ M₁)
(f : Module.End R₁ M₁)
:
def
BilinForm.Nondegenerate
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
(B : BilinForm R M)
:
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
- BilinForm.Nondegenerate B = ∀ (m : M), (∀ (n : M), B.bilin m n = 0) → m = 0
theorem
BilinForm.not_nondegenerate_zero
(R : Type u_1)
(M : Type u_2)
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[Nontrivial M]
:
In a non-trivial module, zero is not non-degenerate.
theorem
BilinForm.Nondegenerate.ne_zero
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[Nontrivial M]
{B : BilinForm R M}
(h : BilinForm.Nondegenerate B)
:
B ≠ 0
theorem
BilinForm.Nondegenerate.congr
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{M' : Type u_9}
[AddCommMonoid M']
[Module R M']
{B : BilinForm R M}
(e : M ≃ₗ[R] M')
(h : BilinForm.Nondegenerate B)
:
@[simp]
theorem
BilinForm.nondegenerate_congr_iff
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{M' : Type u_9}
[AddCommMonoid M']
[Module R M']
{B : BilinForm R M}
(e : M ≃ₗ[R] M')
:
theorem
BilinForm.nondegenerate_iff_ker_eq_bot
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
:
BilinForm.Nondegenerate B ↔ LinearMap.ker (BilinForm.toLin B) = ⊥
A bilinear form is nondegenerate if and only if it has a trivial kernel.
theorem
BilinForm.Nondegenerate.ker_eq_bot
{R : Type u_1}
{M : Type u_2}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : BilinForm R M}
(h : BilinForm.Nondegenerate B)
:
LinearMap.ker (BilinForm.toLin B) = ⊥
theorem
BilinForm.compLeft_injective
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
(B : BilinForm R₁ M₁)
(b : BilinForm.Nondegenerate B)
:
theorem
BilinForm.isAdjointPair_unique_of_nondegenerate
{R₁ : Type u_3}
{M₁ : Type u_4}
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
(B : BilinForm R₁ M₁)
(b : BilinForm.Nondegenerate B)
(φ : M₁ →ₗ[R₁] M₁)
(ψ₁ : M₁ →ₗ[R₁] M₁)
(ψ₂ : M₁ →ₗ[R₁] M₁)
(hψ₁ : BilinForm.IsAdjointPair B B ψ₁ φ)
(hψ₂ : BilinForm.IsAdjointPair B B ψ₂ φ)
:
ψ₁ = ψ₂
noncomputable def
BilinForm.toDual
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
(B : BilinForm K V)
(b : BilinForm.Nondegenerate B)
:
V ≃ₗ[K] Module.Dual K V
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
- BilinForm.toDual B b = LinearMap.linearEquivOfInjective (BilinForm.toLin B) ⋯ ⋯
theorem
BilinForm.toDual_def
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
{B : BilinForm K V}
(b : BilinForm.Nondegenerate B)
{m : V}
{n : V}
:
((BilinForm.toDual B b) m) n = B.bilin m n
theorem
BilinForm.Nondegenerate.flip
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
{B : BilinForm K V}
(hB : BilinForm.Nondegenerate B)
:
BilinForm.Nondegenerate (BilinForm.flip B)
theorem
BilinForm.nonDegenerateFlip_iff
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
{B : BilinForm K V}
:
BilinForm.Nondegenerate (BilinForm.flip B) ↔ BilinForm.Nondegenerate B
noncomputable def
BilinForm.dualBasis
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
{ι : Type u_10}
[DecidableEq ι]
[Finite ι]
(B : BilinForm K V)
(hB : BilinForm.Nondegenerate 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
- BilinForm.dualBasis B hB b = Basis.map (Basis.dualBasis b) (LinearEquiv.symm (BilinForm.toDual B hB))
@[simp]
theorem
BilinForm.dualBasis_repr_apply
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
{ι : Type u_10}
[DecidableEq ι]
[Finite ι]
(B : BilinForm K V)
(hB : BilinForm.Nondegenerate B)
(b : Basis ι K V)
(x : V)
(i : ι)
:
((BilinForm.dualBasis B hB b).repr x) i = B.bilin x (b i)
theorem
BilinForm.apply_dualBasis_left
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
{ι : Type u_10}
[DecidableEq ι]
[Finite ι]
(B : BilinForm K V)
(hB : BilinForm.Nondegenerate B)
(b : Basis ι K V)
(i : ι)
(j : ι)
:
B.bilin ((BilinForm.dualBasis B hB b) i) (b j) = if j = i then 1 else 0
theorem
BilinForm.apply_dualBasis_right
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
{ι : Type u_10}
[DecidableEq ι]
[Finite ι]
(B : BilinForm K V)
(hB : BilinForm.Nondegenerate B)
(sym : BilinForm.IsSymm B)
(b : Basis ι K V)
(i : ι)
(j : ι)
:
B.bilin (b i) ((BilinForm.dualBasis B hB b) j) = if i = j then 1 else 0
@[simp]
theorem
BilinForm.dualBasis_dualBasis_flip
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
(B : BilinForm K V)
(hB : BilinForm.Nondegenerate B)
{ι : Type u_11}
[Finite ι]
[DecidableEq ι]
(b : Basis ι K V)
:
BilinForm.dualBasis B hB (BilinForm.dualBasis (BilinForm.flip B) ⋯ b) = b
@[simp]
theorem
BilinForm.dualBasis_flip_dualBasis
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
(B : BilinForm K V)
(hB : BilinForm.Nondegenerate B)
{ι : Type u_11}
[Finite ι]
[DecidableEq ι]
[FiniteDimensional K V]
(b : Basis ι K V)
:
BilinForm.dualBasis (BilinForm.flip B) ⋯ (BilinForm.dualBasis B hB b) = b
@[simp]
theorem
BilinForm.dualBasis_dualBasis
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
(B : BilinForm K V)
(hB : BilinForm.Nondegenerate B)
(hB' : BilinForm.IsSymm B)
{ι : Type u_11}
[Finite ι]
[DecidableEq ι]
[FiniteDimensional K V]
(b : Basis ι K V)
:
BilinForm.dualBasis B hB (BilinForm.dualBasis B hB b) = b
noncomputable def
BilinForm.symmCompOfNondegenerate
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
(B₁ : BilinForm K V)
(B₂ : BilinForm K V)
(b₂ : BilinForm.Nondegenerate B₂)
:
Given bilinear forms B₁, B₂ where B₂ is nondegenerate, symmCompOfNondegenerate
is the linear map B₂.toLin⁻¹ ∘ B₁.toLin.
Equations
- BilinForm.symmCompOfNondegenerate B₁ B₂ b₂ = LinearMap.comp (↑(LinearEquiv.symm (BilinForm.toDual B₂ b₂))) (BilinForm.toLin B₁)
theorem
BilinForm.comp_symmCompOfNondegenerate_apply
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
(B₁ : BilinForm K V)
{B₂ : BilinForm K V}
(b₂ : BilinForm.Nondegenerate B₂)
(v : V)
:
(BilinForm.toLin B₂) ((BilinForm.symmCompOfNondegenerate B₁ B₂ b₂) v) = (BilinForm.toLin B₁) v
@[simp]
theorem
BilinForm.symmCompOfNondegenerate_left_apply
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
(B₁ : BilinForm K V)
{B₂ : BilinForm K V}
(b₂ : BilinForm.Nondegenerate B₂)
(v : V)
(w : V)
:
B₂.bilin ((BilinForm.symmCompOfNondegenerate B₁ B₂ b₂) w) v = B₁.bilin w v
noncomputable def
BilinForm.leftAdjointOfNondegenerate
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
(B : BilinForm K V)
(b : BilinForm.Nondegenerate 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
theorem
BilinForm.isAdjointPairLeftAdjointOfNondegenerate
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
(B : BilinForm K V)
(b : BilinForm.Nondegenerate B)
(φ : V →ₗ[K] V)
:
BilinForm.IsAdjointPair B B (BilinForm.leftAdjointOfNondegenerate B b φ) φ
theorem
BilinForm.isAdjointPair_iff_eq_of_nondegenerate
{V : Type u_5}
{K : Type u_6}
[Field K]
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
(B : BilinForm K V)
(b : BilinForm.Nondegenerate B)
(ψ : V →ₗ[K] V)
(φ : V →ₗ[K] V)
:
BilinForm.IsAdjointPair B B ψ φ ↔ ψ = BilinForm.leftAdjointOfNondegenerate B b φ
Given the nondegenerate bilinear form B, the linear map φ has a unique left adjoint given by
BilinForm.leftAdjointOfNondegenerate.