Changing the index type of a matrix

This file concerns the map Matrix.reindex, mapping a m by n matrix to an m' by n' matrix, as long as m ≃ m' and n ≃ n'.

Main definitions

• Matrix.reindexLinearEquiv R A: Matrix.reindex is an R-linear equivalence between A-matrices.
• Matrix.reindexAlgEquiv R: Matrix.reindex is an R-algebra equivalence between R-matrices.

Tags

matrix, reindex

def Matrix.reindexLinearEquiv {m : Type u_2} {n : Type u_3} {m' : Type u_6} {n' : Type u_7} (R : Type u_11) (A : Type u_12) [Semiring R] [AddCommMonoid A] [Module R A] (eₘ : m ≃ m') (eₙ : n ≃ n') : Matrix m n A ≃ₗ[R] Matrix m' n' A

The natural map that reindexes a matrix's rows and columns with equivalent types, Matrix.reindex, is a linear equivalence.

Equations

• Matrix.reindexLinearEquiv R A eₘ eₙ = { toFun := (Matrix.reindex eₘ eₙ).toFun, map_add' := ⋯, map_smul' := ⋯, invFun := (Matrix.reindex eₘ eₙ).invFun, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Matrix.reindexLinearEquiv_apply {m : Type u_2} {n : Type u_3} {m' : Type u_6} {n' : Type u_7} (R : Type u_11) (A : Type u_12) [Semiring R] [AddCommMonoid A] [Module R A] (eₘ : m ≃ m') (eₙ : n ≃ n') (M : Matrix m n A) : (Matrix.reindexLinearEquiv R A eₘ eₙ) M = (Matrix.reindex eₘ eₙ) M

@[simp]

theorem Matrix.reindexLinearEquiv_symm {m : Type u_2} {n : Type u_3} {m' : Type u_6} {n' : Type u_7} (R : Type u_11) (A : Type u_12) [Semiring R] [AddCommMonoid A] [Module R A] (eₘ : m ≃ m') (eₙ : n ≃ n') : (Matrix.reindexLinearEquiv R A eₘ eₙ).symm = Matrix.reindexLinearEquiv R A eₘ.symm eₙ.symm

@[simp]

theorem Matrix.reindexLinearEquiv_refl_refl {m : Type u_2} {n : Type u_3} (R : Type u_11) (A : Type u_12) [Semiring R] [AddCommMonoid A] [Module R A] : Matrix.reindexLinearEquiv R A (Equiv.refl m) (Equiv.refl n) = LinearEquiv.refl R (Matrix m n A)

theorem Matrix.reindexLinearEquiv_trans {m : Type u_2} {n : Type u_3} {m' : Type u_6} {n' : Type u_7} {m'' : Type u_9} {n'' : Type u_10} (R : Type u_11) (A : Type u_12) [Semiring R] [AddCommMonoid A] [Module R A] (e₁ : m ≃ m') (e₂ : n ≃ n') (e₁' : m' ≃ m'') (e₂' : n' ≃ n'') : Matrix.reindexLinearEquiv R A e₁ e₂ ≪≫ₗ Matrix.reindexLinearEquiv R A e₁' e₂' = Matrix.reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂')

theorem Matrix.reindexLinearEquiv_comp {m : Type u_2} {n : Type u_3} {m' : Type u_6} {n' : Type u_7} {m'' : Type u_9} {n'' : Type u_10} (R : Type u_11) (A : Type u_12) [Semiring R] [AddCommMonoid A] [Module R A] (e₁ : m ≃ m') (e₂ : n ≃ n') (e₁' : m' ≃ m'') (e₂' : n' ≃ n'') : ⇑(Matrix.reindexLinearEquiv R A e₁' e₂') ∘ ⇑(Matrix.reindexLinearEquiv R A e₁ e₂) = ⇑(Matrix.reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂'))

theorem Matrix.reindexLinearEquiv_comp_apply {m : Type u_2} {n : Type u_3} {m' : Type u_6} {n' : Type u_7} {m'' : Type u_9} {n'' : Type u_10} (R : Type u_11) (A : Type u_12) [Semiring R] [AddCommMonoid A] [Module R A] (e₁ : m ≃ m') (e₂ : n ≃ n') (e₁' : m' ≃ m'') (e₂' : n' ≃ n'') (M : Matrix m n A) : (Matrix.reindexLinearEquiv R A e₁' e₂') ((Matrix.reindexLinearEquiv R A e₁ e₂) M) = (Matrix.reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂')) M

theorem Matrix.reindexLinearEquiv_one {m : Type u_2} {m' : Type u_6} (R : Type u_11) (A : Type u_12) [Semiring R] [AddCommMonoid A] [Module R A] [DecidableEq m] [DecidableEq m'] [One A] (e : m ≃ m') : (Matrix.reindexLinearEquiv R A e e) 1 = 1

theorem Matrix.reindexLinearEquiv_mul {m : Type u_2} {n : Type u_3} {o : Type u_4} {m' : Type u_6} {n' : Type u_7} {o' : Type u_8} (R : Type u_11) (A : Type u_12) [Semiring R] [Semiring A] [Module R A] [Fintype n] [Fintype n'] (eₘ : m ≃ m') (eₙ : n ≃ n') (eₒ : o ≃ o') (M : Matrix m n A) (N : Matrix n o A) : (Matrix.reindexLinearEquiv R A eₘ eₙ) M * (Matrix.reindexLinearEquiv R A eₙ eₒ) N = (Matrix.reindexLinearEquiv R A eₘ eₒ) (M * N)

theorem Matrix.mul_reindexLinearEquiv_one {m : Type u_2} {n : Type u_3} {o : Type u_4} {n' : Type u_7} (R : Type u_11) (A : Type u_12) [Semiring R] [Semiring A] [Module R A] [Fintype n] [DecidableEq o] (e₁ : o ≃ n) (e₂ : o ≃ n') (M : Matrix m n A) : M * (Matrix.reindexLinearEquiv R A e₁ e₂) 1 = (Matrix.reindexLinearEquiv R A (Equiv.refl m) (e₁.symm.trans e₂)) M

def Matrix.reindexAlgEquiv {m : Type u_2} {n : Type u_3} (R : Type u_11) (A : Type u_12) [CommSemiring R] [Fintype n] [Fintype m] [DecidableEq m] [DecidableEq n] [Semiring A] [Algebra R A] (e : m ≃ n) : Matrix m m A ≃ₐ[R] Matrix n n A

For square matrices with coefficients in an algebra over a commutative semiring, the natural map that reindexes a matrix's rows and columns with equivalent types, Matrix.reindex, is an equivalence of algebras.

Equations

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

@[simp]

theorem Matrix.reindexAlgEquiv_apply {m : Type u_2} {n : Type u_3} (R : Type u_11) (A : Type u_12) [CommSemiring R] [Fintype n] [Fintype m] [DecidableEq m] [DecidableEq n] [Semiring A] [Algebra R A] (e : m ≃ n) (M : Matrix m m A) : (Matrix.reindexAlgEquiv R A e) M = (Matrix.reindex e e) M

@[simp]

theorem Matrix.reindexAlgEquiv_symm {m : Type u_2} {n : Type u_3} (R : Type u_11) (A : Type u_12) [CommSemiring R] [Fintype n] [Fintype m] [DecidableEq m] [DecidableEq n] [Semiring A] [Algebra R A] (e : m ≃ n) : (Matrix.reindexAlgEquiv R A e).symm = Matrix.reindexAlgEquiv R A e.symm

@[simp]

theorem Matrix.reindexAlgEquiv_refl {m : Type u_2} (R : Type u_11) (A : Type u_12) [CommSemiring R] [Fintype m] [DecidableEq m] [Semiring A] [Algebra R A] : Matrix.reindexAlgEquiv R A (Equiv.refl m) = AlgEquiv.refl

theorem Matrix.reindexAlgEquiv_mul {m : Type u_2} {n : Type u_3} (R : Type u_11) (A : Type u_12) [CommSemiring R] [Fintype n] [Fintype m] [DecidableEq m] [DecidableEq n] [Semiring A] [Algebra R A] (e : m ≃ n) (M N : Matrix m m A) : (Matrix.reindexAlgEquiv R A e) (M * N) = (Matrix.reindexAlgEquiv R A e) M * (Matrix.reindexAlgEquiv R A e) N

theorem Matrix.det_reindexLinearEquiv_self {m : Type u_2} {n : Type u_3} (R : Type u_11) [CommRing R] [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] (e : m ≃ n) (M : Matrix m m R) : ((Matrix.reindexLinearEquiv R R e e) M).det = M.det

Reindexing both indices along the same equivalence preserves the determinant.

For the simp version of this lemma, see det_submatrix_equiv_self.

theorem Matrix.det_reindexAlgEquiv {m : Type u_2} {n : Type u_3} (R : Type u_11) (B : Type u_13) [CommRing R] [CommRing B] [Algebra R B] [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] (e : m ≃ n) (A : Matrix m m B) : ((Matrix.reindexAlgEquiv R B e) A).det = A.det

Reindexing both indices along the same equivalence preserves the determinant.

For the simp version of this lemma, see det_submatrix_equiv_self.