Bilinear form #

This file defines the conversion between bilinear forms and matrices.

Main definitions #

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

Notations #

In this file we use the following type variables:

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

Tags #

bilinear form, bilin form, BilinearForm, matrix, basis

source

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

BilinForm R₂ (n → R₂)

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

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

Equations

One or more equations did not get rendered due to their size.

Instances For

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theorem Matrix.toBilin'Aux_stdBasis {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (M : Matrix n n R₂) (i : n) (j : n) :

BilinForm.bilin (Matrix.toBilin'Aux M) (↑(LinearMap.stdBasis R₂ (fun x => R₂) i) 1) (↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1) = M i j

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def BilinForm.toMatrixAux {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} (b : n → M₂) :

BilinForm R₂ M₂ →ₗ[R₂] Matrix n n R₂

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

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

Equations

One or more equations did not get rendered due to their size.

Instances For

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@[simp]

theorem BilinForm.toMatrixAux_apply {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} (B : BilinForm R₂ M₂) (b : n → M₂) (i : n) (j : n) :

↑(BilinForm.toMatrixAux b) B i j = BilinForm.bilin B (b i) (b j)

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theorem toBilin'Aux_toMatrixAux {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (B₂ : BilinForm R₂ (n → R₂)) :

Matrix.toBilin'Aux (↑(BilinForm.toMatrixAux fun j => ↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1) B₂) = B₂

`ToMatrix'` section #

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

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def BilinForm.toMatrix' {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] :

BilinForm R₂ (n → R₂) ≃ₗ[R₂] Matrix n n R₂

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

Equations

One or more equations did not get rendered due to their size.

Instances For

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@[simp]

theorem BilinForm.toMatrixAux_stdBasis {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (B : BilinForm R₂ (n → R₂)) :

↑(BilinForm.toMatrixAux fun j => ↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1) B = ↑BilinForm.toMatrix' B

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def Matrix.toBilin' {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] :

Matrix n n R₂ ≃ₗ[R₂] BilinForm R₂ (n → R₂)

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

Equations

Matrix.toBilin' = LinearEquiv.symm BilinForm.toMatrix'

Instances For

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@[simp]

theorem Matrix.toBilin'Aux_eq {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (M : Matrix n n R₂) :

Matrix.toBilin'Aux M = ↑Matrix.toBilin' M

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theorem Matrix.toBilin'_apply {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (M : Matrix n n R₂) (x : n → R₂) (y : n → R₂) :

BilinForm.bilin (↑Matrix.toBilin' M) x y = Finset.sum Finset.univ fun i => Finset.sum Finset.univ fun j => x i * M i j * y j

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theorem Matrix.toBilin'_apply' {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (M : Matrix n n R₂) (v : n → R₂) (w : n → R₂) :

BilinForm.bilin (↑Matrix.toBilin' M) v w = Matrix.dotProduct v (Matrix.mulVec M w)

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@[simp]

theorem Matrix.toBilin'_stdBasis {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (M : Matrix n n R₂) (i : n) (j : n) :

BilinForm.bilin (↑Matrix.toBilin' M) (↑(LinearMap.stdBasis R₂ (fun x => R₂) i) 1) (↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1) = M i j

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@[simp]

theorem BilinForm.toMatrix'_symm {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] :

LinearEquiv.symm BilinForm.toMatrix' = Matrix.toBilin'

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@[simp]

theorem Matrix.toBilin'_symm {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] :

LinearEquiv.symm Matrix.toBilin' = BilinForm.toMatrix'

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@[simp]

theorem Matrix.toBilin'_toMatrix' {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (B : BilinForm R₂ (n → R₂)) :

↑Matrix.toBilin' (↑BilinForm.toMatrix' B) = B

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@[simp]

theorem BilinForm.toMatrix'_toBilin' {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (M : Matrix n n R₂) :

↑BilinForm.toMatrix' (↑Matrix.toBilin' M) = M

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@[simp]

theorem BilinForm.toMatrix'_apply {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (B : BilinForm R₂ (n → R₂)) (i : n) (j : n) :

↑BilinForm.toMatrix' B i j = BilinForm.bilin B (↑(LinearMap.stdBasis R₂ (fun x => R₂) i) 1) (↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1)

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@[simp]

theorem BilinForm.toMatrix'_comp {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} {o : Type u_12} [Fintype n] [Fintype o] [DecidableEq n] [DecidableEq o] (B : BilinForm R₂ (n → R₂)) (l : (o → R₂) →ₗ[R₂] n → R₂) (r : (o → R₂) →ₗ[R₂] n → R₂) :

↑BilinForm.toMatrix' (BilinForm.comp B l r) = Matrix.transpose (↑LinearMap.toMatrix' l) * ↑BilinForm.toMatrix' B * ↑LinearMap.toMatrix' r

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theorem BilinForm.toMatrix'_compLeft {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (B : BilinForm R₂ (n → R₂)) (f : (n → R₂) →ₗ[R₂] n → R₂) :

↑BilinForm.toMatrix' (BilinForm.compLeft B f) = Matrix.transpose (↑LinearMap.toMatrix' f) * ↑BilinForm.toMatrix' B

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theorem BilinForm.toMatrix'_compRight {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (B : BilinForm R₂ (n → R₂)) (f : (n → R₂) →ₗ[R₂] n → R₂) :

↑BilinForm.toMatrix' (BilinForm.compRight B f) = ↑BilinForm.toMatrix' B * ↑LinearMap.toMatrix' f

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theorem BilinForm.mul_toMatrix'_mul {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} {o : Type u_12} [Fintype n] [Fintype o] [DecidableEq n] [DecidableEq o] (B : BilinForm R₂ (n → R₂)) (M : Matrix o n R₂) (N : Matrix n o R₂) :

M * ↑BilinForm.toMatrix' B * N = ↑BilinForm.toMatrix' (BilinForm.comp B (↑Matrix.toLin' (Matrix.transpose M)) (↑Matrix.toLin' N))

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theorem BilinForm.mul_toMatrix' {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (B : BilinForm R₂ (n → R₂)) (M : Matrix n n R₂) :

M * ↑BilinForm.toMatrix' B = ↑BilinForm.toMatrix' (BilinForm.compLeft B (↑Matrix.toLin' (Matrix.transpose M)))

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theorem BilinForm.toMatrix'_mul {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] (B : BilinForm R₂ (n → R₂)) (M : Matrix n n R₂) :

↑BilinForm.toMatrix' B * M = ↑BilinForm.toMatrix' (BilinForm.compRight B (↑Matrix.toLin' M))

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theorem Matrix.toBilin'_comp {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} {o : Type u_12} [Fintype n] [Fintype o] [DecidableEq n] [DecidableEq o] (M : Matrix n n R₂) (P : Matrix n o R₂) (Q : Matrix n o R₂) :

BilinForm.comp (↑Matrix.toBilin' M) (↑Matrix.toLin' P) (↑Matrix.toLin' Q) = ↑Matrix.toBilin' (Matrix.transpose P * M * Q)

`ToMatrix` section #

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

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noncomputable def BilinForm.toMatrix {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) :

BilinForm R₂ M₂ ≃ₗ[R₂] Matrix n n R₂

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

Equations

BilinForm.toMatrix b = LinearEquiv.trans (BilinForm.congr (Basis.equivFun b)) BilinForm.toMatrix'

Instances For

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noncomputable def Matrix.toBilin {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) :

Matrix n n R₂ ≃ₗ[R₂] BilinForm R₂ M₂

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

Equations

Matrix.toBilin b = LinearEquiv.symm (BilinForm.toMatrix b)

Instances For

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@[simp]

theorem BilinForm.toMatrix_apply {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) (i : n) (j : n) :

↑(BilinForm.toMatrix b) B i j = BilinForm.bilin B (↑b i) (↑b j)

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@[simp]

theorem Matrix.toBilin_apply {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) (M : Matrix n n R₂) (x : M₂) (y : M₂) :

BilinForm.bilin (↑(Matrix.toBilin b) M) x y = Finset.sum Finset.univ fun i => Finset.sum Finset.univ fun j => ↑(↑b.repr x) i * M i j * ↑(↑b.repr y) j

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theorem BilinearForm.toMatrixAux_eq {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) :

↑(BilinForm.toMatrixAux ↑b) B = ↑(BilinForm.toMatrix b) B

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@[simp]

theorem BilinForm.toMatrix_symm {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) :

LinearEquiv.symm (BilinForm.toMatrix b) = Matrix.toBilin b

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@[simp]

theorem Matrix.toBilin_symm {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) :

LinearEquiv.symm (Matrix.toBilin b) = BilinForm.toMatrix b

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theorem Matrix.toBilin_basisFun {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] :

Matrix.toBilin (Pi.basisFun R₂ n) = Matrix.toBilin'

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theorem BilinForm.toMatrix_basisFun {R₂ : Type u_5} [CommSemiring R₂] {n : Type u_11} [Fintype n] [DecidableEq n] :

BilinForm.toMatrix (Pi.basisFun R₂ n) = BilinForm.toMatrix'

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@[simp]

theorem Matrix.toBilin_toMatrix {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) :

↑(Matrix.toBilin b) (↑(BilinForm.toMatrix b) B) = B

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@[simp]

theorem BilinForm.toMatrix_toBilin {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) (M : Matrix n n R₂) :

↑(BilinForm.toMatrix b) (↑(Matrix.toBilin b) M) = M

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theorem BilinForm.toMatrix_comp {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} {o : Type u_12} [Fintype n] [Fintype o] [DecidableEq n] (b : Basis n R₂ M₂) {M₂' : Type u_13} [AddCommMonoid M₂'] [Module R₂ M₂'] (c : Basis o R₂ M₂') [DecidableEq o] (B : BilinForm R₂ M₂) (l : M₂' →ₗ[R₂] M₂) (r : M₂' →ₗ[R₂] M₂) :

↑(BilinForm.toMatrix c) (BilinForm.comp B l r) = Matrix.transpose (↑(LinearMap.toMatrix c b) l) * ↑(BilinForm.toMatrix b) B * ↑(LinearMap.toMatrix c b) r

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theorem BilinForm.toMatrix_compLeft {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) (f : M₂ →ₗ[R₂] M₂) :

↑(BilinForm.toMatrix b) (BilinForm.compLeft B f) = Matrix.transpose (↑(LinearMap.toMatrix b b) f) * ↑(BilinForm.toMatrix b) B

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theorem BilinForm.toMatrix_compRight {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) (f : M₂ →ₗ[R₂] M₂) :

↑(BilinForm.toMatrix b) (BilinForm.compRight B f) = ↑(BilinForm.toMatrix b) B * ↑(LinearMap.toMatrix b b) f

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@[simp]

theorem BilinForm.toMatrix_mul_basis_toMatrix {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} {o : Type u_12} [Fintype n] [Fintype o] [DecidableEq n] (b : Basis n R₂ M₂) [DecidableEq o] (c : Basis o R₂ M₂) (B : BilinForm R₂ M₂) :

Matrix.transpose (Basis.toMatrix b ↑c) * ↑(BilinForm.toMatrix b) B * Basis.toMatrix b ↑c = ↑(BilinForm.toMatrix c) B

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theorem BilinForm.mul_toMatrix_mul {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} {o : Type u_12} [Fintype n] [Fintype o] [DecidableEq n] (b : Basis n R₂ M₂) {M₂' : Type u_13} [AddCommMonoid M₂'] [Module R₂ M₂'] (c : Basis o R₂ M₂') [DecidableEq o] (B : BilinForm R₂ M₂) (M : Matrix o n R₂) (N : Matrix n o R₂) :

M * ↑(BilinForm.toMatrix b) B * N = ↑(BilinForm.toMatrix c) (BilinForm.comp B (↑(Matrix.toLin c b) (Matrix.transpose M)) (↑(Matrix.toLin c b) N))

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theorem BilinForm.mul_toMatrix {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) (M : Matrix n n R₂) :

M * ↑(BilinForm.toMatrix b) B = ↑(BilinForm.toMatrix b) (BilinForm.compLeft B (↑(Matrix.toLin b b) (Matrix.transpose M)))

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theorem BilinForm.toMatrix_mul {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) (B : BilinForm R₂ M₂) (M : Matrix n n R₂) :

↑(BilinForm.toMatrix b) B * M = ↑(BilinForm.toMatrix b) (BilinForm.compRight B (↑(Matrix.toLin b b) M))

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theorem Matrix.toBilin_comp {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} {o : Type u_12} [Fintype n] [Fintype o] [DecidableEq n] (b : Basis n R₂ M₂) {M₂' : Type u_13} [AddCommMonoid M₂'] [Module R₂ M₂'] (c : Basis o R₂ M₂') [DecidableEq o] (M : Matrix n n R₂) (P : Matrix n o R₂) (Q : Matrix n o R₂) :

BilinForm.comp (↑(Matrix.toBilin b) M) (↑(Matrix.toLin c b) P) (↑(Matrix.toLin c b) Q) = ↑(Matrix.toBilin c) (Matrix.transpose P * M * Q)

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@[simp]

theorem isAdjointPair_toBilin' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J : Matrix n n R₃) (J₃ : Matrix n n R₃) (A : Matrix n n R₃) (A' : Matrix n n R₃) [DecidableEq n] :

BilinForm.IsAdjointPair (↑Matrix.toBilin' J) (↑Matrix.toBilin' J₃) (↑Matrix.toLin' A) (↑Matrix.toLin' A') ↔ Matrix.IsAdjointPair J J₃ A A'

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@[simp]

theorem isAdjointPair_toBilin {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [AddCommGroup M₃] [Module R₃ M₃] {n : Type u_11} [Fintype n] (b : Basis n R₃ M₃) (J : Matrix n n R₃) (J₃ : Matrix n n R₃) (A : Matrix n n R₃) (A' : Matrix n n R₃) [DecidableEq n] :

BilinForm.IsAdjointPair (↑(Matrix.toBilin b) J) (↑(Matrix.toBilin b) J₃) (↑(Matrix.toLin b b) A) (↑(Matrix.toLin b b) A') ↔ Matrix.IsAdjointPair J J₃ A A'

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theorem Matrix.isAdjointPair_equiv' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J : Matrix n n R₃) (A : Matrix n n R₃) (A' : Matrix n n R₃) [DecidableEq n] (P : Matrix n n R₃) (h : IsUnit P) :

Matrix.IsAdjointPair (Matrix.transpose P * J * P) (Matrix.transpose P * J * P) A A' ↔ Matrix.IsAdjointPair J J (P * A * P⁻¹) (P * A' * P⁻¹)

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def pairSelfAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J : Matrix n n R₃) (J₃ : Matrix n n R₃) [DecidableEq n] :

Submodule R₃ (Matrix n n R₃)

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

Equations

pairSelfAdjointMatricesSubmodule' J J₃ = Submodule.map (↑LinearMap.toMatrix') (BilinForm.isPairSelfAdjointSubmodule (↑Matrix.toBilin' J) (↑Matrix.toBilin' J₃))

Instances For

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theorem mem_pairSelfAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J : Matrix n n R₃) (J₃ : Matrix n n R₃) (A : Matrix n n R₃) [DecidableEq n] :

A ∈ pairSelfAdjointMatricesSubmodule J J₃ ↔ Matrix.IsAdjointPair J J₃ A A

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def selfAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J : Matrix n n R₃) [DecidableEq n] :

Submodule R₃ (Matrix n n R₃)

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

Equations

selfAdjointMatricesSubmodule' J = pairSelfAdjointMatricesSubmodule J J

Instances For

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theorem mem_selfAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J : Matrix n n R₃) (A : Matrix n n R₃) [DecidableEq n] :

A ∈ selfAdjointMatricesSubmodule J ↔ Matrix.IsSelfAdjoint J A

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def skewAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J : Matrix n n R₃) [DecidableEq n] :

Submodule R₃ (Matrix n n R₃)

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

Equations

skewAdjointMatricesSubmodule' J = pairSelfAdjointMatricesSubmodule (-J) J

Instances For

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theorem mem_skewAdjointMatricesSubmodule' {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J : Matrix n n R₃) (A : Matrix n n R₃) [DecidableEq n] :

A ∈ skewAdjointMatricesSubmodule J ↔ Matrix.IsSkewAdjoint J A

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theorem Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι R₂} (b : Basis ι R₂ M₂) :

BilinForm.Nondegenerate (↑Matrix.toBilin' M) ↔ BilinForm.Nondegenerate (↑(Matrix.toBilin b) M)

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theorem Matrix.Nondegenerate.toBilin' {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι R₃} (h : Matrix.Nondegenerate M) :

BilinForm.Nondegenerate (↑Matrix.toBilin' M)

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@[simp]

theorem Matrix.nondegenerate_toBilin'_iff {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι R₃} :

BilinForm.Nondegenerate (↑Matrix.toBilin' M) ↔ Matrix.Nondegenerate M

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theorem Matrix.Nondegenerate.toBilin {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [AddCommGroup M₃] [Module R₃ M₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι R₃} (h : Matrix.Nondegenerate M) (b : Basis ι R₃ M₃) :

BilinForm.Nondegenerate (↑(Matrix.toBilin b) M)

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@[simp]

theorem Matrix.nondegenerate_toBilin_iff {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [AddCommGroup M₃] [Module R₃ M₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι R₃} (b : Basis ι R₃ M₃) :

BilinForm.Nondegenerate (↑(Matrix.toBilin b) M) ↔ Matrix.Nondegenerate M

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@[simp]

theorem BilinForm.nondegenerate_toMatrix'_iff {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {B : BilinForm R₃ (ι → R₃)} :

Matrix.Nondegenerate (↑BilinForm.toMatrix' B) ↔ BilinForm.Nondegenerate B

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theorem BilinForm.Nondegenerate.toMatrix' {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {B : BilinForm R₃ (ι → R₃)} (h : BilinForm.Nondegenerate B) :

Matrix.Nondegenerate (↑BilinForm.toMatrix' B)

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@[simp]

theorem BilinForm.nondegenerate_toMatrix_iff {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [AddCommGroup M₃] [Module R₃ M₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {B : BilinForm R₃ M₃} (b : Basis ι R₃ M₃) :

Matrix.Nondegenerate (↑(BilinForm.toMatrix b) B) ↔ BilinForm.Nondegenerate B

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theorem BilinForm.Nondegenerate.toMatrix {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [AddCommGroup M₃] [Module R₃ M₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {B : BilinForm R₃ M₃} (h : BilinForm.Nondegenerate B) (b : Basis ι R₃ M₃) :

Matrix.Nondegenerate (↑(BilinForm.toMatrix b) B)

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theorem BilinForm.nondegenerate_toBilin'_iff_det_ne_zero {A : Type u_11} [CommRing A] [IsDomain A] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι A} :

BilinForm.Nondegenerate (↑Matrix.toBilin' M) ↔ Matrix.det M ≠ 0

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theorem BilinForm.nondegenerate_toBilin'_of_det_ne_zero' {A : Type u_11} [CommRing A] [IsDomain A] {ι : Type u_12} [DecidableEq ι] [Fintype ι] (M : Matrix ι ι A) (h : Matrix.det M ≠ 0) :

BilinForm.Nondegenerate (↑Matrix.toBilin' M)

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theorem BilinForm.nondegenerate_iff_det_ne_zero {M₃ : Type u_8} [AddCommGroup M₃] {A : Type u_11} [CommRing A] [IsDomain A] [Module A M₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {B : BilinForm A M₃} (b : Basis ι A M₃) :

BilinForm.Nondegenerate B ↔ Matrix.det (↑(BilinForm.toMatrix b) B) ≠ 0

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theorem BilinForm.nondegenerate_of_det_ne_zero {M₃ : Type u_8} [AddCommGroup M₃] {A : Type u_11} [CommRing A] [IsDomain A] [Module A M₃] (B₃ : BilinForm A M₃) {ι : Type u_12} [DecidableEq ι] [Fintype ι] (b : Basis ι A M₃) (h : Matrix.det (↑(BilinForm.toMatrix b) B₃) ≠ 0) :

BilinForm.Nondegenerate B₃

Bilinear form #

Main definitions #

Notations #

Tags #

ToMatrix' section #

ToMatrix section #

`ToMatrix'` section #

`ToMatrix` section #