Sesquilinear maps #
This files provides properties about sesquilinear maps and forms. The maps considered are of the
form M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M, where I₁ : R₁ →+* R and I₂ : R₂ →+* R are ring homomorphisms and
M₁ is a module over R₁, M₂ is a module over R₂ and M is a module over R.
Sesquilinear forms are the special case that M₁ = M₂, M = R₁ = R₂ = R, and I₁ = RingHom.id R.
Taking additionally I₂ = RingHom.id R, then one obtains bilinear forms.
Sesquilinear maps are a special case of the bilinear maps defined in BilinearMap.lean and many
basic lemmas about construction and elementary calculations are found there.
Main declarations #
IsOrtho: states that two vectors are orthogonal with respect to a sesquilinear mapIsSymm,IsAlt: states that a sesquilinear form is symmetric and alternating, respectivelyorthogonalBilin: provides the orthogonal complement with respect to sesquilinear form
References #
Tags #
Sesquilinear form, Sesquilinear map,
Orthogonal vectors #
def
LinearMap.IsOrtho
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
[AddCommMonoid M]
[Module R M]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
(B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M)
(x : M₁)
(y : M₂)
:
The proposition that two elements of a sesquilinear map space are orthogonal
Instances For
theorem
LinearMap.isOrtho_def
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
[AddCommMonoid M]
[Module R M]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
{B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M}
{x : M₁}
{y : M₂}
:
theorem
LinearMap.isOrtho_zero_left
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
[AddCommMonoid M]
[Module R M]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
(B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M)
(x : M₂)
:
B.IsOrtho 0 x
theorem
LinearMap.isOrtho_zero_right
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[CommSemiring R₂]
[AddCommMonoid M₂]
[Module R₂ M₂]
[AddCommMonoid M]
[Module R M]
{I₁ : R₁ →+* R}
{I₂ : R₂ →+* R}
(B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M)
(x : M₁)
:
B.IsOrtho x 0
theorem
LinearMap.isOrtho_flip
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[AddCommMonoid M]
[Module R M]
{I₁ I₁' : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M}
{x y : M₁}
:
B.IsOrtho x y ↔ B.flip.IsOrtho y x
def
LinearMap.IsOrthoᵢ
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
{n : Type u_19}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[AddCommMonoid M]
[Module R M]
{I₁ I₁' : R₁ →+* R}
(B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M)
(v : n → M₁)
:
A set of vectors v is orthogonal with respect to some bilinear map B if and only
if for all i ≠ j, B (v i) (v j) = 0. For orthogonality between two elements, use
BilinForm.isOrtho
Instances For
theorem
LinearMap.isOrthoᵢ_def
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
{n : Type u_19}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[AddCommMonoid M]
[Module R M]
{I₁ I₁' : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M}
{v : n → M₁}
:
theorem
LinearMap.isOrthoᵢ_flip
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
{n : Type u_19}
[CommSemiring R]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
[AddCommMonoid M]
[Module R M]
{I₁ I₁' : R₁ →+* R}
(B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M)
{v : n → M₁}
:
B.IsOrthoᵢ v ↔ B.flip.IsOrthoᵢ v
theorem
LinearMap.ortho_smul_left
{K : Type u_13}
{K₁ : Type u_14}
{K₂ : Type u_15}
{V : Type u_16}
{V₁ : Type u_17}
{V₂ : Type u_18}
[Field K]
[AddCommGroup V]
[Module K V]
[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] V}
{x : V₁}
{y : V₂}
{a : K₁}
(ha : a ≠ 0)
:
theorem
LinearMap.ortho_smul_right
{K : Type u_13}
{K₁ : Type u_14}
{K₂ : Type u_15}
{V : Type u_16}
{V₁ : Type u_17}
{V₂ : Type u_18}
[Field K]
[AddCommGroup V]
[Module K V]
[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] V}
{x : V₁}
{y : V₂}
{a : K₂}
{ha : a ≠ 0}
:
theorem
LinearMap.linearIndependent_of_isOrthoᵢ
{K : Type u_13}
{K₁ : Type u_14}
{V : Type u_16}
{V₁ : Type u_17}
{n : Type u_19}
[Field K]
[AddCommGroup V]
[Module K V]
[Field K₁]
[AddCommGroup V₁]
[Module K₁ V₁]
{I₁ I₁' : K₁ →+* K}
{B : V₁ →ₛₗ[I₁] V₁ →ₛₗ[I₁'] V}
{v : n → V₁}
(hv₁ : B.IsOrthoᵢ v)
(hv₂ : ∀ (i : n), ¬B.IsOrtho (v i) (v i))
:
LinearIndependent K₁ v
A set of orthogonal vectors v with respect to some sesquilinear map B is linearly
independent if for all i, B (v i) (v i) ≠ 0.
Reflexive bilinear maps #
def
LinearMap.IsRefl
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
(B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M)
:
The proposition that a sesquilinear map is reflexive
Instances For
theorem
LinearMap.IsRefl.eq_zero
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
(H : B.IsRefl)
{x y : M₁}
:
theorem
LinearMap.IsRefl.eq_iff
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
(H : B.IsRefl)
{x y : M₁}
:
theorem
LinearMap.IsRefl.ortho_comm
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
(H : B.IsRefl)
{x y : M₁}
:
B.IsOrtho x y ↔ B.IsOrtho y x
theorem
LinearMap.IsRefl.domRestrict
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
(H : B.IsRefl)
(p : Submodule R₁ M₁)
:
(B.domRestrict₁₂ p p).IsRefl
@[simp]
theorem
LinearMap.IsRefl.flip_isRefl_iff
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
:
B.flip.IsRefl ↔ B.IsRefl
theorem
LinearMap.IsRefl.ker_flip_eq_bot
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
(H : B.IsRefl)
(h : LinearMap.ker B = ⊥)
:
LinearMap.ker B.flip = ⊥
theorem
LinearMap.IsRefl.ker_eq_bot_iff_ker_flip_eq_bot
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
(H : B.IsRefl)
:
LinearMap.ker B = ⊥ ↔ LinearMap.ker B.flip = ⊥
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
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 : B.IsSymm)
(x 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 : B.IsSymm)
:
B.IsRefl
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 : B.IsSymm)
{x y : M}
:
B.IsOrtho x y ↔ B.IsOrtho 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 : B.IsSymm)
(p : Submodule R M)
:
(B.domRestrict₁₂ p p).IsSymm
@[simp]
theorem
LinearMap.isSymm_zero
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{I : R →+* R}
:
theorem
LinearMap.isSymm_iff_eq_flip
{R : Type u_1}
{M : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
{B : LinearMap.BilinForm R M}
:
LinearMap.IsSymm B ↔ B = LinearMap.flip B
Alternating bilinear maps #
def
LinearMap.IsAlt
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
(B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M)
:
The proposition that a sesquilinear map is alternating
Instances For
theorem
LinearMap.IsAlt.self_eq_zero
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
(H : B.IsAlt)
(x : M₁)
:
(B x) x = 0
theorem
LinearMap.IsAlt.eq_of_add_add_eq_zero
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
(H : B.IsAlt)
[IsCancelAdd M]
{a b c : M₁}
(hAdd : a + b + c = 0)
:
(B a) b = (B b) c
theorem
LinearMap.IsAlt.neg
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommGroup M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
(H : B.IsAlt)
(x y : M₁)
:
theorem
LinearMap.IsAlt.isRefl
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommGroup M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
(H : B.IsAlt)
:
B.IsRefl
theorem
LinearMap.IsAlt.ortho_comm
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommSemiring R]
[AddCommGroup M]
[Module R M]
[CommSemiring R₁]
[AddCommMonoid M₁]
[Module R₁ M₁]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
(H : B.IsAlt)
{x y : M₁}
:
B.IsOrtho x y ↔ B.IsOrtho 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}
:
The orthogonal complement #
def
Submodule.orthogonalBilin
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommRing R]
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
[AddCommGroup M]
[Module R M]
{I₁ I₂ : R₁ →+* R}
(N : Submodule R₁ M₁)
(B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M)
:
Submodule R₁ M₁
The orthogonal complement of a submodule N with respect to some bilinear map 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 maps 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
- N.orthogonalBilin B = { carrier := {m : M₁ | ∀ n ∈ N, B.IsOrtho n m}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
Instances For
theorem
Submodule.orthogonalBilin_le
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommRing R]
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
[AddCommGroup M]
[Module R M]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
{N L : Submodule R₁ M₁}
(h : N ≤ L)
:
L.orthogonalBilin B ≤ N.orthogonalBilin B
theorem
Submodule.le_orthogonalBilin_orthogonalBilin
{R : Type u_1}
{R₁ : Type u_2}
{M : Type u_5}
{M₁ : Type u_6}
[CommRing R]
[CommRing R₁]
[AddCommGroup M₁]
[Module R₁ M₁]
[AddCommGroup M]
[Module R M]
{I₁ I₂ : R₁ →+* R}
{B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}
{N : Submodule R₁ M₁}
(b : B.IsRefl)
:
N ≤ (N.orthogonalBilin B).orthogonalBilin B
theorem
LinearMap.span_singleton_inf_orthogonal_eq_bot
{K : Type u_13}
{K₁ : Type u_14}
{V₁ : Type u_17}
{V₂ : Type u_18}
[Field K]
[Field K₁]
[AddCommGroup V₁]
[Module K₁ V₁]
[AddCommGroup V₂]
[Module K V₂]
{J₁ J₁' : K₁ →+* K}
(B : V₁ →ₛₗ[J₁] V₁ →ₛₗ[J₁'] V₂)
(x : V₁)
(hx : ¬B.IsOrtho x x)
:
Submodule.span K₁ {x} ⊓ (Submodule.span K₁ {x}).orthogonalBilin B = ⊥
theorem
LinearMap.orthogonal_span_singleton_eq_to_lin_ker
{K : Type u_13}
{V : Type u_16}
{V₂ : Type u_18}
[Field K]
[AddCommGroup V]
[Module K V]
[AddCommGroup V₂]
[Module K V₂]
{J : K →+* K}
{B : V →ₗ[K] V →ₛₗ[J] V₂}
(x : V)
:
(Submodule.span K {x}).orthogonalBilin B = LinearMap.ker (B x)
theorem
LinearMap.span_singleton_sup_orthogonal_eq_top
{K : Type u_13}
{V : Type u_16}
[Field K]
[AddCommGroup V]
[Module K V]
{B : V →ₗ[K] V →ₗ[K] K}
{x : V}
(hx : ¬B.IsOrtho x x)
:
Submodule.span K {x} ⊔ (Submodule.span K {x}).orthogonalBilin B = ⊤
theorem
LinearMap.isCompl_span_singleton_orthogonal
{K : Type u_13}
{V : Type u_16}
[Field K]
[AddCommGroup V]
[Module K V]
{B : V →ₗ[K] V →ₗ[K] K}
{x : V}
(hx : ¬B.IsOrtho x x)
:
IsCompl (Submodule.span K {x}) ((Submodule.span K {x}).orthogonalBilin 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}
{M₃ : Type u_8}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₁]
[Module R M₁]
[AddCommMonoid M₃]
[Module R M₃]
{I : R →+* R}
(B : M →ₗ[R] M →ₛₗ[I] M₃)
(B' : M₁ →ₗ[R] M₁ →ₛₗ[I] M₃)
(f : M → M₁)
(g : M₁ → M)
:
Given a pair of modules equipped with bilinear maps, this is the condition for a pair of maps between them to be mutually adjoint.
Instances For
theorem
LinearMap.isAdjointPair_iff_comp_eq_compl₂
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
{M₃ : Type u_8}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₁]
[Module R M₁]
[AddCommMonoid M₃]
[Module R M₃]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] M₃}
{B' : M₁ →ₗ[R] M₁ →ₛₗ[I] M₃}
{f : M →ₗ[R] M₁}
{g : M₁ →ₗ[R] M}
:
theorem
LinearMap.isAdjointPair_zero
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
{M₃ : Type u_8}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₁]
[Module R M₁]
[AddCommMonoid M₃]
[Module R M₃]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] M₃}
{B' : M₁ →ₗ[R] M₁ →ₛₗ[I] M₃}
:
B.IsAdjointPair B' 0 0
theorem
LinearMap.isAdjointPair_id
{R : Type u_1}
{M : Type u_5}
{M₃ : Type u_8}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₃]
[Module R M₃]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] M₃}
:
B.IsAdjointPair B id id
theorem
LinearMap.isAdjointPair_one
{R : Type u_1}
{M : Type u_5}
{M₃ : Type u_8}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₃]
[Module R M₃]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] M₃}
:
B.IsAdjointPair B ⇑1 ⇑1
theorem
LinearMap.IsAdjointPair.add
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
{M₃ : Type u_8}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₁]
[Module R M₁]
[AddCommMonoid M₃]
[Module R M₃]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] M₃}
{B' : M₁ →ₗ[R] M₁ →ₛₗ[I] M₃}
{f f' : M → M₁}
{g g' : M₁ → M}
(h : B.IsAdjointPair B' f g)
(h' : B.IsAdjointPair B' f' g')
:
theorem
LinearMap.IsAdjointPair.comp
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
{M₃ : Type u_8}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₁]
[Module R M₁]
[AddCommMonoid M₂]
[Module R M₂]
[AddCommMonoid M₃]
[Module R M₃]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] M₃}
{B' : M₁ →ₗ[R] M₁ →ₛₗ[I] M₃}
{B'' : M₂ →ₗ[R] M₂ →ₛₗ[I] M₃}
{f : M → M₁}
{g : M₁ → M}
{f' : M₁ → M₂}
{g' : M₂ → M₁}
(h : B.IsAdjointPair B' f g)
(h' : B'.IsAdjointPair B'' f' g')
:
theorem
LinearMap.IsAdjointPair.mul
{R : Type u_1}
{M : Type u_5}
{M₃ : Type u_8}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid M₃]
[Module R M₃]
{I : R →+* R}
{B : M →ₗ[R] M →ₛₗ[I] M₃}
{f g f' g' : Module.End R M}
(h : B.IsAdjointPair B ⇑f ⇑g)
(h' : B.IsAdjointPair B ⇑f' ⇑g')
:
theorem
LinearMap.IsAdjointPair.sub
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₁]
[Module R M₁]
[AddCommGroup M₂]
[Module R M₂]
{B : M →ₗ[R] M →ₗ[R] M₂}
{B' : M₁ →ₗ[R] M₁ →ₗ[R] M₂}
{f f' : M → M₁}
{g g' : M₁ → M}
(h : B.IsAdjointPair B' f g)
(h' : B.IsAdjointPair B' f' g')
:
theorem
LinearMap.IsAdjointPair.smul
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₁]
[Module R M₁]
[AddCommGroup M₂]
[Module R M₂]
{B : M →ₗ[R] M →ₗ[R] M₂}
{B' : M₁ →ₗ[R] M₁ →ₗ[R] M₂}
{f : M → M₁}
{g : M₁ → M}
(c : R)
(h : B.IsAdjointPair B' f g)
:
def
LinearMap.IsOrthogonal
{R : Type u_20}
{M : Type u_21}
[CommRing R]
[AddCommGroup M]
[Module R M]
(B : LinearMap.BilinForm R M)
(f : M → M)
:
A linear transformation f is orthogonal with respect to a bilinear form B if B is
bi-invariant with respect to f.
Equations
- LinearMap.IsOrthogonal B f = ∀ (x y : M), (B (f x)) (f y) = (B x) y
Instances For
@[simp]
theorem
LinearEquiv.isAdjointPair_symm_iff
{R : Type u_20}
{M : Type u_21}
[CommRing R]
[AddCommGroup M]
[Module R M]
{B : LinearMap.BilinForm R M}
{f : M ≃ M}
:
LinearMap.IsAdjointPair B B ⇑f ⇑f.symm ↔ LinearMap.IsOrthogonal B ⇑f
theorem
LinearMap.isOrthogonal_of_forall_apply_same
{R : Type u_20}
{M : Type u_21}
[CommRing R]
[AddCommGroup M]
[Module R M]
{B : LinearMap.BilinForm R M}
{F : Type u_22}
[FunLike F M M]
[LinearMapClass F R M M]
(f : F)
(h : IsLeftRegular 2)
(hB : LinearMap.IsSymm B)
(hf : ∀ (x : M), (B (f x)) (f x) = (B x) x)
:
Self-adjoint pairs #
def
LinearMap.IsPairSelfAdjoint
{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 F : M →ₗ[R] M →ₛₗ[I] M₁)
(f : M → M)
:
The condition for an endomorphism to be "self-adjoint" with respect to a pair of bilinear maps on the underlying module. In the case that these two maps are identical, this is the usual concept of self adjointness. In the case that one of the maps is the negation of the other, this is the usual concept of skew adjointness.
Equations
- B.IsPairSelfAdjoint F f = B.IsAdjointPair F f f
Instances For
def
LinearMap.IsSelfAdjoint
{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] M₁)
(f : M → M)
:
An endomorphism of a module is self-adjoint with respect to a bilinear map if it serves as an adjoint for itself.
Equations
- B.IsSelfAdjoint f = B.IsAdjointPair B f f
Instances For
def
LinearMap.isPairSelfAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
{M₂ : Type u_7}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₂]
[Module R M₂]
(B F : M →ₗ[R] M →ₗ[R] M₂)
:
Submodule R (Module.End R M)
The set of pair-self-adjoint endomorphisms are a submodule of the type of all endomorphisms.
Equations
- B.isPairSelfAdjointSubmodule F = { carrier := {f : Module.End R M | B.IsPairSelfAdjoint F ⇑f}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
Instances For
def
LinearMap.IsSkewAdjoint
{R : Type u_1}
{M : Type u_5}
{M₂ : Type u_7}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₂]
[Module R M₂]
(B : M →ₗ[R] M →ₗ[R] M₂)
(f : M → M)
:
An endomorphism of a module is skew-adjoint with respect to a bilinear map if its negation serves as an adjoint.
Instances For
def
LinearMap.selfAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
{M₂ : Type u_7}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₂]
[Module R M₂]
(B : M →ₗ[R] M →ₗ[R] M₂)
:
Submodule R (Module.End R M)
The set of self-adjoint endomorphisms of a module with bilinear map is a submodule. (In fact it is a Jordan subalgebra.)
Equations
- B.selfAdjointSubmodule = B.isPairSelfAdjointSubmodule B
Instances For
def
LinearMap.skewAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
{M₂ : Type u_7}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₂]
[Module R M₂]
(B : M →ₗ[R] M →ₗ[R] M₂)
:
Submodule R (Module.End R M)
The set of skew-adjoint endomorphisms of a module with bilinear map is a submodule. (In fact it is a Lie subalgebra.)
Instances For
@[simp]
theorem
LinearMap.mem_isPairSelfAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
{M₂ : Type u_7}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₂]
[Module R M₂]
{B F : M →ₗ[R] M →ₗ[R] M₂}
(f : Module.End R M)
:
theorem
LinearMap.isPairSelfAdjoint_equiv
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₁]
[Module R M₁]
[AddCommGroup M₂]
[Module R M₂]
{B F : M →ₗ[R] M →ₗ[R] M₂}
(e : M₁ ≃ₗ[R] M)
(f : Module.End R M)
:
B.IsPairSelfAdjoint F ⇑f ↔ (B.compl₁₂ ↑e ↑e).IsPairSelfAdjoint (F.compl₁₂ ↑e ↑e) ⇑(e.symm.conj f)
@[simp]
theorem
LinearMap.mem_selfAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
{M₂ : Type u_7}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₂]
[Module R M₂]
{B : M →ₗ[R] M →ₗ[R] M₂}
(f : Module.End R M)
:
@[simp]
theorem
LinearMap.mem_skewAdjointSubmodule
{R : Type u_1}
{M : Type u_5}
{M₂ : Type u_7}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₂]
[Module R M₂]
{B : M →ₗ[R] M →ₗ[R] M₂}
(f : Module.End R M)
:
Nondegenerate bilinear maps #
def
LinearMap.SeparatingLeft
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M)
:
A bilinear map 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.
Instances For
theorem
LinearMap.not_separatingLeft_zero
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
(M₁ : Type u_6)
(M₂ : Type u_7)
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M}
(h : B.SeparatingLeft)
:
B ≠ 0
theorem
LinearMap.SeparatingLeft.congr
{R : Type u_1}
{M : Type u_5}
{Mₗ₁ : Type u_9}
{Mₗ₁' : Type u_10}
{Mₗ₂ : Type u_11}
{Mₗ₂' : Type u_12}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M}
(e₁ : Mₗ₁ ≃ₗ[R] Mₗ₁')
(e₂ : Mₗ₂ ≃ₗ[R] Mₗ₂')
(h : B.SeparatingLeft)
:
((e₁.arrowCongr (e₂.arrowCongr (LinearEquiv.refl R M))) B).SeparatingLeft
@[simp]
theorem
LinearMap.separatingLeft_congr_iff
{R : Type u_1}
{M : Type u_5}
{Mₗ₁ : Type u_9}
{Mₗ₁' : Type u_10}
{Mₗ₂ : Type u_11}
{Mₗ₂' : Type u_12}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M}
(e₁ : Mₗ₁ ≃ₗ[R] Mₗ₁')
(e₂ : Mₗ₂ ≃ₗ[R] Mₗ₂')
:
((e₁.arrowCongr (e₂.arrowCongr (LinearEquiv.refl R M))) B).SeparatingLeft ↔ B.SeparatingLeft
def
LinearMap.SeparatingRight
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M)
:
A bilinear map 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.
Instances For
def
LinearMap.Nondegenerate
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M)
:
A bilinear map is called non-degenerate if it is left-separating and right-separating.
Instances For
@[simp]
theorem
LinearMap.flip_separatingRight
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M}
:
B.flip.SeparatingRight ↔ B.SeparatingLeft
@[simp]
theorem
LinearMap.flip_separatingLeft
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M}
:
B.flip.SeparatingLeft ↔ B.SeparatingRight
@[simp]
theorem
LinearMap.flip_nondegenerate
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M}
:
B.flip.Nondegenerate ↔ B.Nondegenerate
theorem
LinearMap.separatingLeft_iff_linear_nontrivial
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M}
:
theorem
LinearMap.separatingRight_iff_linear_flip_nontrivial
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M}
:
theorem
LinearMap.separatingLeft_iff_ker_eq_bot
{R : Type u_1}
{R₁ : Type u_2}
{R₂ : Type u_3}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M}
:
B.SeparatingLeft ↔ LinearMap.ker B = ⊥
A bilinear map 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}
{M : Type u_5}
{M₁ : Type u_6}
{M₂ : Type u_7}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[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] M}
:
B.SeparatingRight ↔ LinearMap.ker B.flip = ⊥
A bilinear map is right-separating if and only if its flip has a trivial kernel.
theorem
LinearMap.IsRefl.nondegenerate_iff_separatingLeft
{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] M₁}
(hB : B.IsRefl)
:
B.Nondegenerate ↔ B.SeparatingLeft
theorem
LinearMap.IsRefl.nondegenerate_iff_separatingRight
{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] M₁}
(hB : B.IsRefl)
:
B.Nondegenerate ↔ B.SeparatingRight
theorem
LinearMap.disjoint_ker_of_nondegenerate_restrict
{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] M₁}
{W : Submodule R M}
(hW : (B.domRestrict₁₂ W W).Nondegenerate)
:
Disjoint W (LinearMap.ker B)
theorem
LinearMap.IsSymm.nondegenerate_restrict_of_isCompl_ker
{R : Type u_1}
{M : Type u_5}
[CommRing R]
[AddCommGroup M]
[Module R M]
{B : M →ₗ[R] M →ₗ[R] R}
(hB : B.IsSymm)
{W : Submodule R M}
(hW : IsCompl W (LinearMap.ker B))
:
(B.domRestrict₁₂ W W).Nondegenerate
theorem
LinearMap.nondegenerate_restrict_of_disjoint_orthogonal
{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] M₁}
(hB : B.IsRefl)
{W : Submodule R M}
(hW : Disjoint W (W.orthogonalBilin B))
:
(B.domRestrict₁₂ W W).Nondegenerate
The restriction of a reflexive bilinear map 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}
{M₁ : Type u_6}
{n : Type u_19}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₁]
[Module R M₁]
{I I' : R →+* R}
[Nontrivial R]
{B : M →ₛₗ[I] M →ₛₗ[I'] M₁}
{v : Basis n R M}
(h : B.IsOrthoᵢ ⇑v)
(hB : B.SeparatingLeft)
(i : n)
:
¬B.IsOrtho (v i) (v i)
An orthogonal basis with respect to a left-separating bilinear map has no self-orthogonal elements.
theorem
LinearMap.IsOrthoᵢ.not_isOrtho_basis_self_of_separatingRight
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
{n : Type u_19}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₁]
[Module R M₁]
{I I' : R →+* R}
[Nontrivial R]
{B : M →ₛₗ[I] M →ₛₗ[I'] M₁}
{v : Basis n R M}
(h : B.IsOrthoᵢ ⇑v)
(hB : B.SeparatingRight)
(i : n)
:
¬B.IsOrtho (v i) (v i)
An orthogonal basis with respect to a right-separating bilinear map has no self-orthogonal elements.
theorem
LinearMap.IsOrthoᵢ.separatingLeft_of_not_isOrtho_basis_self
{R : Type u_1}
{M : Type u_5}
{M₁ : Type u_6}
{n : Type u_19}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₁]
[Module R M₁]
[NoZeroSMulDivisors R M₁]
{B : M →ₗ[R] M →ₗ[R] M₁}
(v : Basis n R M)
(hO : B.IsOrthoᵢ ⇑v)
(h : ∀ (i : n), ¬B.IsOrtho (v i) (v i))
:
B.SeparatingLeft
Given an orthogonal basis with respect to a bilinear map, the bilinear map 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}
{M₁ : Type u_6}
{n : Type u_19}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₁]
[Module R M₁]
[NoZeroSMulDivisors R M₁]
{B : M →ₗ[R] M →ₗ[R] M₁}
(v : Basis n R M)
(hO : B.IsOrthoᵢ ⇑v)
(h : ∀ (i : n), ¬B.IsOrtho (v i) (v i))
:
B.SeparatingRight
Given an orthogonal basis with respect to a bilinear map, the bilinear map 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}
{M₁ : Type u_6}
{n : Type u_19}
[CommRing R]
[AddCommGroup M]
[Module R M]
[AddCommGroup M₁]
[Module R M₁]
[NoZeroSMulDivisors R M₁]
{B : M →ₗ[R] M →ₗ[R] M₁}
(v : Basis n R M)
(hO : B.IsOrthoᵢ ⇑v)
(h : ∀ (i : n), ¬B.IsOrtho (v i) (v i))
:
B.Nondegenerate
Given an orthogonal basis with respect to a bilinear map, the bilinear map is nondegenerate if the basis has no elements which are self-orthogonal.
theorem
LinearMap.BilinForm.apply_mul_apply_le_of_forall_zero_le
{R : Type u_1}
{M : Type u_5}
[LinearOrderedCommRing R]
[AddCommGroup M]
[Module R M]
(B : LinearMap.BilinForm R M)
(hs : ∀ (x : M), 0 ≤ (B x) x)
(x y : M)
:
The Cauchy-Schwarz inequality for positive semidefinite forms.
theorem
LinearMap.BilinForm.apply_sq_le_of_symm
{R : Type u_1}
{M : Type u_5}
[LinearOrderedCommRing R]
[AddCommGroup M]
[Module R M]
(B : LinearMap.BilinForm R M)
(hs : ∀ (x : M), 0 ≤ (B x) x)
(hB : LinearMap.IsSymm B)
(x y : M)
:
The Cauchy-Schwarz inequality for positive semidefinite symmetric forms.
theorem
LinearMap.BilinForm.not_linearIndependent_of_apply_mul_apply_eq
{R : Type u_1}
{M : Type u_5}
[LinearOrderedCommRing R]
[AddCommGroup M]
[Module R M]
(B : LinearMap.BilinForm R M)
(hp : ∀ (x : M), x ≠ 0 → 0 < (B x) x)
(x y : M)
(he : (B x) y * (B y) x = (B x) x * (B y) y)
:
¬LinearIndependent R ![x, y]
The equality case of Cauchy-Schwarz.
theorem
LinearMap.BilinForm.apply_mul_apply_lt_iff_linearIndependent
{R : Type u_1}
{M : Type u_5}
[LinearOrderedCommRing R]
[AddCommGroup M]
[Module R M]
(B : LinearMap.BilinForm R M)
[NoZeroSMulDivisors R M]
(hp : ∀ (x : M), x ≠ 0 → 0 < (B x) x)
(x y : M)
:
(B x) y * (B y) x < (B x) x * (B y) y ↔ LinearIndependent R ![x, y]
Strict Cauchy-Schwarz is equivalent to linear independence for positive definite forms.
theorem
LinearMap.BilinForm.apply_sq_lt_iff_linearIndependent_of_symm
{R : Type u_1}
{M : Type u_5}
[LinearOrderedCommRing R]
[AddCommGroup M]
[Module R M]
(B : LinearMap.BilinForm R M)
[NoZeroSMulDivisors R M]
(hp : ∀ (x : M), x ≠ 0 → 0 < (B x) x)
(hB : LinearMap.IsSymm B)
(x y : M)
:
(B x) y ^ 2 < (B x) x * (B y) y ↔ LinearIndependent R ![x, y]
Strict Cauchy-Schwarz is equivalent to linear independence for positive definite symmetric forms.
theorem
LinearMap.BilinForm.apply_apply_same_eq_zero_iff
{R : Type u_1}
{M : Type u_5}
[LinearOrderedCommRing R]
[AddCommGroup M]
[Module R M]
(B : LinearMap.BilinForm R M)
(hs : ∀ (x : M), 0 ≤ (B x) x)
(hB : LinearMap.IsSymm B)
{x : M}
:
(B x) x = 0 ↔ x ∈ LinearMap.ker B
theorem
LinearMap.BilinForm.nondegenerate_iff
{R : Type u_1}
{M : Type u_5}
[LinearOrderedCommRing R]
[AddCommGroup M]
[Module R M]
(B : LinearMap.BilinForm R M)
(hs : ∀ (x : M), 0 ≤ (B x) x)
(hB : LinearMap.IsSymm B)
:
LinearMap.Nondegenerate B ↔ ∀ (x : M), (B x) x = 0 ↔ x = 0
theorem
LinearMap.BilinForm.nondegenerate_iff'
{R : Type u_1}
{M : Type u_5}
[LinearOrderedCommRing R]
[AddCommGroup M]
[Module R M]
(B : LinearMap.BilinForm R M)
(hs : ∀ (x : M), 0 ≤ (B x) x)
(hB : LinearMap.IsSymm B)
:
LinearMap.Nondegenerate B ↔ ∀ (x : M), x ≠ 0 → 0 < (B x) x
A convenience variant of LinearMap.BilinForm.nondegenerate_iff characterising nondegeneracy as
positive definiteness.
theorem
LinearMap.BilinForm.nondegenerate_restrict_iff_disjoint_ker
{R : Type u_1}
{M : Type u_5}
[LinearOrderedCommRing R]
[AddCommGroup M]
[Module R M]
(B : LinearMap.BilinForm R M)
(hs : ∀ (x : M), 0 ≤ (B x) x)
(hB : LinearMap.IsSymm B)
{W : Submodule R M}
:
(LinearMap.domRestrict₁₂ B W W).Nondegenerate ↔ Disjoint W (LinearMap.ker B)