Sesquilinear form #
This files provides properties about sesquilinear forms. The maps considered are of the form
M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R, where I₁ : R₁ →+* R and I₂ : R₂ →+* R are ring homomorphisms and
M₁ is a module over R₁ and M₂ is a module over R₂.
Sesquilinear forms are the special case that M₁ = M₂, R₁ = R₂ = R, and I₁ = RingHom.id R.
Taking additionally I₂ = RingHom.id R, then one obtains bilinear forms.
These forms are a special case of the bilinear maps defined in BilinearMap.lean and all basic
lemmas about construction and elementary calculations are found there.
Main declarations #
IsOrtho: states that two vectors are orthogonal with respect to a sesquilinear formIsSymm,IsAlt: states that a sesquilinear form is symmetric and alternating, respectivelyorthogonalBilin: provides the orthogonal complement with respect to sesquilinear form
References #
Tags #
Sesquilinear form,
Orthogonal vectors #
def
LinearMap.IsOrtho
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
(B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R)
(x : M₁)
(y : M₂)
:
The proposition that two elements of a sesquilinear form space are orthogonal
Equations
- LinearMap.IsOrtho B x y = (↑(↑B x) y = 0)
Instances For
theorem
LinearMap.isOrtho_def
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
{B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R}
{x : M₁}
{y : M₂}
:
LinearMap.IsOrtho B x y ↔ ↑(↑B x) y = 0
theorem
LinearMap.isOrtho_zero_left
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
(B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R)
(x : M₂)
:
LinearMap.IsOrtho B 0 x
theorem
LinearMap.isOrtho_zero_right
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
(B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R)
(x : M₁)
:
LinearMap.IsOrtho B x 0
theorem
LinearMap.isOrtho_flip
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₁' : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] R}
{x : M₁}
{y : M₁}
:
LinearMap.IsOrtho B x y ↔ LinearMap.IsOrtho (LinearMap.flip B) y x
def
LinearMap.IsOrthoᵢ
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
{n : Type u_18}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₁' : R₁ →+* R}
(B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] R)
(v : n → M₁)
:
A set of vectors v is orthogonal with respect to some bilinear form B if and only
if for all i ≠ j, B (v i) (v j) = 0. For orthogonality between two elements, use
BilinForm.isOrtho
Equations
- LinearMap.IsOrthoᵢ B v = Pairwise (LinearMap.IsOrtho B on v)
Instances For
theorem
LinearMap.isOrthoᵢ_def
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
{n : Type u_18}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₁' : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] R}
{v : n → M₁}
:
LinearMap.IsOrthoᵢ B v ↔ ∀ (i j : n), i ≠ j → ↑(↑B (v i)) (v j) = 0
theorem
LinearMap.isOrthoᵢ_flip
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
{n : Type u_18}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₁' : R₁ →+* R}
(B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] R)
{v : n → M₁}
:
LinearMap.IsOrthoᵢ B v ↔ LinearMap.IsOrthoᵢ (LinearMap.flip B) v
theorem
LinearMap.ortho_smul_left
{K : Type u_12}
{K₁ : Type u_13}
{K₂ : Type u_14}
{V₁ : Type u_16}
{V₂ : Type u_17}
[Field K]
[Field K₁]
[AddCommGroup V₁]
[Module K₁ V₁]
[Field K₂]
[AddCommGroup V₂]
[Module K₂ V₂]
{I₁ : K₁ →+* K}
{I₂ : K₂ →+* K}
{B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K}
{x : V₁}
{y : V₂}
{a : K₁}
(ha : a ≠ 0)
:
LinearMap.IsOrtho B x y ↔ LinearMap.IsOrtho B (a • x) y
theorem
LinearMap.ortho_smul_right
{K : Type u_12}
{K₁ : Type u_13}
{K₂ : Type u_14}
{V₁ : Type u_16}
{V₂ : Type u_17}
[Field K]
[Field K₁]
[AddCommGroup V₁]
[Module K₁ V₁]
[Field K₂]
[AddCommGroup V₂]
[Module K₂ V₂]
{I₁ : K₁ →+* K}
{I₂ : K₂ →+* K}
{B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] K}
{x : V₁}
{y : V₂}
{a : K₂}
{ha : a ≠ 0}
:
LinearMap.IsOrtho B x y ↔ LinearMap.IsOrtho B x (a • y)
theorem
LinearMap.linearIndependent_of_isOrthoᵢ
{K : Type u_12}
{K₁ : Type u_13}
{V₁ : Type u_16}
{n : Type u_18}
[Field K]
[Field K₁]
[AddCommGroup V₁]
[Module K₁ V₁]
{I₁ : K₁ →+* K}
{I₁' : K₁ →+* K}
{B : V₁ →ₛₗ[I₁] V₁ →ₛₗ[I₁'] K}
{v : n → V₁}
(hv₁ : LinearMap.IsOrthoᵢ B v)
(hv₂ : ∀ (i : n), ¬LinearMap.IsOrtho B (v i) (v i))
:
LinearIndependent K₁ v
A set of orthogonal vectors v with respect to some sesquilinear form B is linearly
independent if for all i, B (v i) (v i) ≠ 0.
Reflexive bilinear forms #
def
LinearMap.IsRefl
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
(B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R)
:
The proposition that a sesquilinear form is reflexive
Equations
- LinearMap.IsRefl B = ∀ (x y : M₁), ↑(↑B x) y = 0 → ↑(↑B y) x = 0
Instances For
theorem
LinearMap.IsRefl.eq_zero
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R}
(H : LinearMap.IsRefl B)
{x : M₁}
{y : M₁}
:
theorem
LinearMap.IsRefl.ortho_comm
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R}
(H : LinearMap.IsRefl B)
{x : M₁}
{y : M₁}
:
LinearMap.IsOrtho B x y ↔ LinearMap.IsOrtho B y x
theorem
LinearMap.IsRefl.domRestrict
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R}
(H : LinearMap.IsRefl B)
(p : Submodule R₁ M₁)
:
@[simp]
theorem
LinearMap.IsRefl.flip_isRefl_iff
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R}
:
theorem
LinearMap.IsRefl.ker_flip_eq_bot
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R}
(H : LinearMap.IsRefl B)
(h : LinearMap.ker B = ⊥)
:
theorem
LinearMap.IsRefl.ker_eq_bot_iff_ker_flip_eq_bot
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R}
(H : LinearMap.IsRefl B)
:
LinearMap.ker B = ⊥ ↔ LinearMap.ker (LinearMap.flip B) = ⊥
Symmetric bilinear forms #
def
LinearMap.IsSymm
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{I : R →+* R}
(B : M →ₛₗ[I] M →ₗ[R] R)
:
The proposition that a sesquilinear form is symmetric
Equations
- LinearMap.IsSymm B = ∀ (x y : M), ↑I (↑(↑B x) y) = ↑(↑B y) x
Instances For
theorem
LinearMap.IsSymm.eq
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{I : R →+* R}
{B : M →ₛₗ[I] M →ₗ[R] R}
(H : LinearMap.IsSymm B)
(x : M)
(y : M)
:
↑I (↑(↑B x) y) = ↑(↑B y) x
theorem
LinearMap.IsSymm.isRefl
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{I : R →+* R}
{B : M →ₛₗ[I] M →ₗ[R] R}
(H : LinearMap.IsSymm B)
:
theorem
LinearMap.IsSymm.ortho_comm
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{I : R →+* R}
{B : M →ₛₗ[I] M →ₗ[R] R}
(H : LinearMap.IsSymm B)
{x : M}
{y : M}
:
LinearMap.IsOrtho B x y ↔ LinearMap.IsOrtho B y x
theorem
LinearMap.IsSymm.domRestrict
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{I : R →+* R}
{B : M →ₛₗ[I] M →ₗ[R] R}
(H : LinearMap.IsSymm B)
(p : Submodule R M)
:
theorem
LinearMap.isSymm_iff_eq_flip
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : M →ₗ[R] M →ₗ[R] R}
:
LinearMap.IsSymm B ↔ B = LinearMap.flip B
Alternating bilinear forms #
def
LinearMap.IsAlt
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommRing R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
(B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R)
:
The proposition that a sesquilinear form is alternating
Equations
- LinearMap.IsAlt B = ∀ (x : M₁), ↑(↑B x) x = 0
Instances For
theorem
LinearMap.IsAlt.self_eq_zero
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommRing R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R}
(H : LinearMap.IsAlt B)
(x : M₁)
:
↑(↑B x) x = 0
theorem
LinearMap.IsAlt.neg
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommRing R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R}
(H : LinearMap.IsAlt B)
(x : M₁)
(y : M₁)
:
theorem
LinearMap.IsAlt.isRefl
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommRing R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R}
(H : LinearMap.IsAlt B)
:
theorem
LinearMap.IsAlt.ortho_comm
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommRing R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R}
(H : LinearMap.IsAlt B)
{x : M₁}
{y : M₁}
:
LinearMap.IsOrtho B x y ↔ LinearMap.IsOrtho B y x
theorem
LinearMap.isAlt_iff_eq_neg_flip
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommRing R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I : R₁ →+* R}
[NoZeroDivisors R]
[CharZero R]
{B : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R}
:
LinearMap.IsAlt B ↔ B = -LinearMap.flip B
The orthogonal complement #
def
Submodule.orthogonalBilin
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommRing R]
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
(N : Submodule R₁ M₁)
(B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R)
:
Submodule R₁ M₁
The orthogonal complement of a submodule N with respect to some bilinear form is the set of
elements x which are orthogonal to all elements of N; i.e., for all y in N, B x y = 0.
Note that for general (neither symmetric nor antisymmetric) bilinear forms this definition has a
chirality; in addition to this "left" orthogonal complement one could define a "right" orthogonal
complement for which, for all y in N, B y x = 0. This variant definition is not currently
provided in mathlib.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem
Submodule.mem_orthogonalBilin_iff
{R : Type u_1}
{R₁ : Type u_2}
{M₁ : Type u_6}
[CommRing R]
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
{I₁ : R₁ →+* R}
{I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R}
{N : Submodule R₁ M₁}
{m : M₁}
:
m ∈ Submodule.orthogonalBilin N B ↔ ∀ (n : M₁), n ∈ N → LinearMap.IsOrtho B n m
theorem
LinearMap.span_singleton_inf_orthogonal_eq_bot
{K : Type u_12}
{K₁ : Type u_13}
{V₁ : Type u_16}
[Field K]
[Field K₁]
[AddCommGroup V₁]
[Module K₁ V₁]
{J₁ : K₁ →+* K}
{J₁' : K₁ →+* K}
(B : V₁ →ₛₗ[J₁] V₁ →ₛₗ[J₁'] K)
(x : V₁)
(hx : ¬LinearMap.IsOrtho B x x)
:
Submodule.span K₁ {x} ⊓ Submodule.orthogonalBilin (Submodule.span K₁ {x}) B = ⊥
theorem
LinearMap.orthogonal_span_singleton_eq_to_lin_ker
{K : Type u_12}
{V : Type u_15}
[Field K]
[AddCommGroup V]
[Module K V]
{J : K →+* K}
{B : V →ₗ[K] V →ₛₗ[J] K}
(x : V)
:
Submodule.orthogonalBilin (Submodule.span K {x}) B = LinearMap.ker (↑B x)
theorem
LinearMap.span_singleton_sup_orthogonal_eq_top
{K : Type u_12}
{V : Type u_15}
[Field K]
[AddCommGroup V]
[Module K V]
{B : V →ₗ[K] V →ₗ[K] K}
{x : V}
(hx : ¬LinearMap.IsOrtho B x x)
:
Submodule.span K {x} ⊔ Submodule.orthogonalBilin (Submodule.span K {x}) B = ⊤
theorem
LinearMap.isCompl_span_singleton_orthogonal
{K : Type u_12}
{V : Type u_15}
[Field K]
[AddCommGroup V]
[Module K V]
{B : V →ₗ[K] V →ₗ[K] K}
{x : V}
(hx : ¬LinearMap.IsOrtho B x x)
:
IsCompl (Submodule.span K {x}) (Submodule.orthogonalBilin (Submodule.span K {x}) B)
Given a bilinear form B and some x such that B x x ≠ 0, the span of the singleton of x
is complement to its orthogonal complement.
Adjoint pairs #
def
LinearMap.IsAdjointPair
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₁]
[Module R M₁]
{I : R →+* R}
(B : M →ₗ[R] M →ₛₗ[I] R)
(B' : M₁ →ₗ[R] M₁ →ₛₗ[I] R)
(f : M →ₗ[R] M₁)
(g : M₁ →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
- LinearMap.IsAdjointPair B B' f g = ∀ (x : M) (y : M₁), ↑(↑B' (↑f x)) y = ↑(↑B x) (↑g y)
Instances For
theorem
LinearMap.isAdjointPair_iff_comp_eq_compl₂
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₁]
[Module R M₁]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] R}
{B' : M₁ →ₗ[R] M₁ →ₛₗ[I] R}
{f : M →ₗ[R] M₁}
{g : M₁ →ₗ[R] M}
:
LinearMap.IsAdjointPair B B' f g ↔ LinearMap.comp B' f = LinearMap.compl₂ B g
theorem
LinearMap.isAdjointPair_zero
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₁]
[Module R M₁]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] R}
{B' : M₁ →ₗ[R] M₁ →ₛₗ[I] R}
:
LinearMap.IsAdjointPair B B' 0 0
theorem
LinearMap.isAdjointPair_id
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] R}
:
LinearMap.IsAdjointPair B B 1 1
theorem
LinearMap.IsAdjointPair.add
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₁]
[Module R M₁]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] R}
{B' : M₁ →ₗ[R] M₁ →ₛₗ[I] R}
{f : M →ₗ[R] M₁}
{f' : M →ₗ[R] M₁}
{g : M₁ →ₗ[R] M}
{g' : M₁ →ₗ[R] M}
(h : LinearMap.IsAdjointPair B B' f g)
(h' : LinearMap.IsAdjointPair B B' f' g')
:
LinearMap.IsAdjointPair B B' (f + f') (g + g')
theorem
LinearMap.IsAdjointPair.comp
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₁]
[Module R M₁]
[AddCommMonoid M₂]
[Module R M₂]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] R}
{B' : M₁ →ₗ[R] M₁ →ₛₗ[I] R}
{B'' : M₂ →ₗ[R] M₂ →ₛₗ[I] R}
{f : M →ₗ[R] M₁}
{g : M₁ →ₗ[R] M}
{f' : M₁ →ₗ[R] M₂}
{g' : M₂ →ₗ[R] M₁}
(h : LinearMap.IsAdjointPair B B' f g)
(h' : LinearMap.IsAdjointPair B' B'' f' g')
:
LinearMap.IsAdjointPair B B'' (LinearMap.comp f' f) (LinearMap.comp g g')
theorem
LinearMap.IsAdjointPair.mul
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] R}
{f : Module.End R M}
{g : Module.End R M}
{f' : Module.End R M}
{g' : Module.End R M}
(h : LinearMap.IsAdjointPair B B f g)
(h' : LinearMap.IsAdjointPair B B f' g')
:
LinearMap.IsAdjointPair B B (f * f') (g' * g)
theorem
LinearMap.IsAdjointPair.sub
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₁]
[Module R M₁]
{B : M →ₗ[R] M →ₗ[R] R}
{B' : M₁ →ₗ[R] M₁ →ₗ[R] R}
{f : M →ₗ[R] M₁}
{f' : M →ₗ[R] M₁}
{g : M₁ →ₗ[R] M}
{g' : M₁ →ₗ[R] M}
(h : LinearMap.IsAdjointPair B B' f g)
(h' : LinearMap.IsAdjointPair B B' f' g')
:
LinearMap.IsAdjointPair B B' (f - f') (g - g')
theorem
LinearMap.IsAdjointPair.smul
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₁]
[Module R M₁]
{B : M →ₗ[R] M →ₗ[R] R}
{B' : M₁ →ₗ[R] M₁ →ₗ[R] R}
{f : M →ₗ[R] M₁}
{g : M₁ →ₗ[R] M}
(c : R)
(h : LinearMap.IsAdjointPair B B' f g)
:
LinearMap.IsAdjointPair B B' (c • f) (c • g)
Self-adjoint pairs #
def
LinearMap.IsPairSelfAdjoint
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{I : R →+* R}
(B : M →ₗ[R] M →ₛₗ[I] R)
(F : M →ₗ[R] M →ₛₗ[I] R)
(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
- LinearMap.IsPairSelfAdjoint B F f = LinearMap.IsAdjointPair B F f f
Instances For
def
LinearMap.IsSelfAdjoint
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{I : R →+* R}
(B : M →ₗ[R] M →ₛₗ[I] R)
(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
- LinearMap.IsSelfAdjoint B f = LinearMap.IsAdjointPair B B f f
Instances For
def
LinearMap.isPairSelfAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
(B : M →ₗ[R] M →ₗ[R] R)
(F : M →ₗ[R] M →ₗ[R] R)
:
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.
Instances For
def
LinearMap.IsSkewAdjoint
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
(B : M →ₗ[R] M →ₗ[R] R)
(f : Module.End R M)
:
An endomorphism of a module is skew-adjoint with respect to a bilinear form if its negation serves as an adjoint.
Equations
- LinearMap.IsSkewAdjoint B f = LinearMap.IsAdjointPair B B f (-f)
Instances For
def
LinearMap.selfAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
(B : M →ₗ[R] M →ₗ[R] R)
:
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
Instances For
def
LinearMap.skewAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
(B : M →ₗ[R] M →ₗ[R] R)
:
Submodule R (Module.End R M)
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.mem_isPairSelfAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
{B : M →ₗ[R] M →ₗ[R] R}
{F : M →ₗ[R] M →ₗ[R] R}
(f : Module.End R M)
:
theorem
LinearMap.isPairSelfAdjoint_equiv
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₁]
[Module R M₁]
{B : M →ₗ[R] M →ₗ[R] R}
{F : M →ₗ[R] M →ₗ[R] R}
(e : M₁ ≃ₗ[R] M)
(f : Module.End R M)
:
LinearMap.IsPairSelfAdjoint B F f ↔ LinearMap.IsPairSelfAdjoint (LinearMap.compl₁₂ B ↑e ↑e) (LinearMap.compl₁₂ F ↑e ↑e)
(↑(LinearEquiv.conj (LinearEquiv.symm e)) f)
theorem
LinearMap.isSkewAdjoint_iff_neg_self_adjoint
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
{B : M →ₗ[R] M →ₗ[R] R}
(f : Module.End R M)
:
LinearMap.IsSkewAdjoint B f ↔ LinearMap.IsAdjointPair (-B) B f f
@[simp]
theorem
LinearMap.mem_selfAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
{B : M →ₗ[R] M →ₗ[R] R}
(f : Module.End R M)
:
@[simp]
theorem
LinearMap.mem_skewAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
{B : M →ₗ[R] M →ₗ[R] R}
(f : Module.End R M)
:
Nondegenerate bilinear forms #
def
LinearMap.SeparatingLeft
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
(B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R)
:
A bilinear form is called left-separating if
the only element that is left-orthogonal to every other element is 0; i.e.,
for every nonzero x in M₁, there exists y in M₂ with B x y ≠ 0.
Equations
- LinearMap.SeparatingLeft B = ∀ (x : M₁), (∀ (y : M₂), ↑(↑B x) y = 0) → x = 0
Instances For
theorem
LinearMap.not_separatingLeft_zero
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
(M₁ : Type u_6)
(M₂ : Type u_7)
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
(I₁ : R₁ →+* R)
(I₂ : R₂ →+* R)
[Nontrivial M₁]
:
In a non-trivial module, zero is not non-degenerate.
theorem
LinearMap.SeparatingLeft.ne_zero
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
[Nontrivial M₁]
{B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R}
(h : LinearMap.SeparatingLeft B)
:
B ≠ 0
theorem
LinearMap.SeparatingLeft.congr
{R : Type u_1}
{Mₗ₁ : Type u_8}
{Mₗ₁' : Type u_9}
{Mₗ₂ : Type u_10}
{Mₗ₂' : Type u_11}
[CommSemiring R]
[AddCommMonoid Mₗ₁]
[AddCommMonoid Mₗ₂]
[AddCommMonoid Mₗ₁']
[AddCommMonoid Mₗ₂']
[Module R Mₗ₁]
[Module R Mₗ₂]
[Module R Mₗ₁']
[Module R Mₗ₂']
{B : Mₗ₁ →ₗ[R] Mₗ₂ →ₗ[R] R}
(e₁ : Mₗ₁ ≃ₗ[R] Mₗ₁')
(e₂ : Mₗ₂ ≃ₗ[R] Mₗ₂')
(h : LinearMap.SeparatingLeft B)
:
LinearMap.SeparatingLeft (↑(LinearEquiv.arrowCongr e₁ (LinearEquiv.arrowCongr e₂ (LinearEquiv.refl R R))) B)
@[simp]
theorem
LinearMap.separatingLeft_congr_iff
{R : Type u_1}
{Mₗ₁ : Type u_8}
{Mₗ₁' : Type u_9}
{Mₗ₂ : Type u_10}
{Mₗ₂' : Type u_11}
[CommSemiring R]
[AddCommMonoid Mₗ₁]
[AddCommMonoid Mₗ₂]
[AddCommMonoid Mₗ₁']
[AddCommMonoid Mₗ₂']
[Module R Mₗ₁]
[Module R Mₗ₂]
[Module R Mₗ₁']
[Module R Mₗ₂']
{B : Mₗ₁ →ₗ[R] Mₗ₂ →ₗ[R] R}
(e₁ : Mₗ₁ ≃ₗ[R] Mₗ₁')
(e₂ : Mₗ₂ ≃ₗ[R] Mₗ₂')
:
LinearMap.SeparatingLeft (↑(LinearEquiv.arrowCongr e₁ (LinearEquiv.arrowCongr e₂ (LinearEquiv.refl R R))) B) ↔ LinearMap.SeparatingLeft B
def
LinearMap.SeparatingRight
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
(B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R)
:
A bilinear form is called right-separating if
the only element that is right-orthogonal to every other element is 0; i.e.,
for every nonzero y in M₂, there exists x in M₁ with B x y ≠ 0.
Equations
- LinearMap.SeparatingRight B = ∀ (y : M₂), (∀ (x : M₁), ↑(↑B x) y = 0) → y = 0
Instances For
def
LinearMap.Nondegenerate
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
(B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R)
:
A bilinear form is called non-degenerate if it is left-separating and right-separating.
Equations
Instances For
@[simp]
theorem
LinearMap.flip_separatingRight
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
{B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R}
:
@[simp]
theorem
LinearMap.flip_separatingLeft
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
{B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R}
:
@[simp]
theorem
LinearMap.flip_nondegenerate
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
{B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R}
:
theorem
LinearMap.separatingLeft_iff_linear_nontrivial
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
{B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R}
:
LinearMap.SeparatingLeft B ↔ ∀ (x : M₁), ↑B x = 0 → x = 0
theorem
LinearMap.separatingRight_iff_linear_flip_nontrivial
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
{B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R}
:
LinearMap.SeparatingRight B ↔ ∀ (y : M₂), ↑(LinearMap.flip B) y = 0 → y = 0
theorem
LinearMap.separatingLeft_iff_ker_eq_bot
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
{B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R}
:
A bilinear form is left-separating if and only if it has a trivial kernel.
theorem
LinearMap.separatingRight_iff_flip_ker_eq_bot
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
{B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R}
:
A bilinear form is right-separating if and only if its flip has a trivial kernel.
theorem
LinearMap.IsRefl.nondegenerate_of_separatingLeft
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
{B : M →ₗ[R] M →ₗ[R] R}
(hB : LinearMap.IsRefl B)
(hB' : LinearMap.SeparatingLeft B)
:
theorem
LinearMap.IsRefl.nondegenerate_of_separatingRight
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
{B : M →ₗ[R] M →ₗ[R] R}
(hB : LinearMap.IsRefl B)
(hB' : LinearMap.SeparatingRight B)
:
theorem
LinearMap.nondegenerateRestrictOfDisjointOrthogonal
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
{B : M →ₗ[R] M →ₗ[R] R}
(hB : LinearMap.IsRefl B)
{W : Submodule R M}
(hW : Disjoint W (Submodule.orthogonalBilin W B))
:
The restriction of a reflexive bilinear form B onto a submodule W is
nondegenerate if W has trivial intersection with its orthogonal complement,
that is Disjoint W (W.orthogonalBilin B).
theorem
LinearMap.IsOrthoᵢ.not_isOrtho_basis_self_of_separatingLeft
{R : Type u_1}
{M : Type u_5}
{n : Type u_18}
[CommRing R]
[AddCommGroup M]
[Module R M]
{I : R →+* R}
{I' : R →+* R}
[Nontrivial R]
{B : M →ₛₗ[I] M →ₛₗ[I'] R}
{v : Basis n R M}
(h : LinearMap.IsOrthoᵢ B ↑v)
(hB : LinearMap.SeparatingLeft B)
(i : n)
:
¬LinearMap.IsOrtho B (↑v i) (↑v i)
An orthogonal basis with respect to a left-separating bilinear form has no self-orthogonal elements.
theorem
LinearMap.IsOrthoᵢ.not_isOrtho_basis_self_of_separatingRight
{R : Type u_1}
{M : Type u_5}
{n : Type u_18}
[CommRing R]
[AddCommGroup M]
[Module R M]
{I : R →+* R}
{I' : R →+* R}
[Nontrivial R]
{B : M →ₛₗ[I] M →ₛₗ[I'] R}
{v : Basis n R M}
(h : LinearMap.IsOrthoᵢ B ↑v)
(hB : LinearMap.SeparatingRight B)
(i : n)
:
¬LinearMap.IsOrtho B (↑v i) (↑v i)
An orthogonal basis with respect to a right-separating bilinear form has no self-orthogonal elements.
theorem
LinearMap.IsOrthoᵢ.separatingLeft_of_not_isOrtho_basis_self
{R : Type u_1}
{M : Type u_5}
{n : Type u_18}
[CommRing R]
[AddCommGroup M]
[Module R M]
[NoZeroDivisors R]
{B : M →ₗ[R] M →ₗ[R] R}
(v : Basis n R M)
(hO : LinearMap.IsOrthoᵢ B ↑v)
(h : ∀ (i : n), ¬LinearMap.IsOrtho B (↑v i) (↑v i))
:
Given an orthogonal basis with respect to a bilinear form, the bilinear form is left-separating if the basis has no elements which are self-orthogonal.
theorem
LinearMap.IsOrthoᵢ.separatingRight_iff_not_isOrtho_basis_self
{R : Type u_1}
{M : Type u_5}
{n : Type u_18}
[CommRing R]
[AddCommGroup M]
[Module R M]
[NoZeroDivisors R]
{B : M →ₗ[R] M →ₗ[R] R}
(v : Basis n R M)
(hO : LinearMap.IsOrthoᵢ B ↑v)
(h : ∀ (i : n), ¬LinearMap.IsOrtho B (↑v i) (↑v i))
:
Given an orthogonal basis with respect to a bilinear form, the bilinear form is right-separating if the basis has no elements which are self-orthogonal.
theorem
LinearMap.IsOrthoᵢ.nondegenerate_of_not_isOrtho_basis_self
{R : Type u_1}
{M : Type u_5}
{n : Type u_18}
[CommRing R]
[AddCommGroup M]
[Module R M]
[NoZeroDivisors R]
{B : M →ₗ[R] M →ₗ[R] R}
(v : Basis n R M)
(hO : LinearMap.IsOrthoᵢ B ↑v)
(h : ∀ (i : n), ¬LinearMap.IsOrtho B (↑v i) (↑v i))
:
Given an orthogonal basis with respect to a bilinear form, the bilinear form is nondegenerate if the basis has no elements which are self-orthogonal.