Composition of matrices

This file shows that Mₙ(Mₘ(R)) ≃ Mₙₘ(R), Mₙ(Rᵒᵖ) ≃ₐ[K] Mₙ(R)ᵒᵖ and also different levels of equivalence when R is an AddCommMonoid, Semiring, and Algebra over a CommSemiring K.

Main definitions

• Matrix.comp is an equivalence between Matrix I J (Matrix K L R) and I × K by J × L matrices.
• Matrix.compAddEquiv: Matrix.comp as an AddEquiv
• Matrix.compRingEquiv: Matrix.comp as a RingEquiv
• Matrix.compLinearEquiv: Matrix.comp as a LinearEquiv
• Matrix.compAlgEquiv: Matrix.comp as an AlgEquiv

def Matrix.comp (I : Type u_1) (J : Type u_2) (K : Type u_3) (L : Type u_4) (R : Type u_5) : Matrix I J (Matrix K L R) ≃ Matrix (I × K) (J × L) R

An I by J matrix where each entry is a K by L matrix is equivalent to an I × K by J × L matrix.

This is available in bundled forms as:

• Matrix.compAddEquiv
• Matrix.compLinearEquiv
• Matrix.compRingEquiv
• Matrix.compAlgEquiv

Equations

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

@[simp]

theorem Matrix.comp_symm_apply (I : Type u_1) (J : Type u_2) (K : Type u_3) (L : Type u_4) (R : Type u_5) (n : Matrix (I × K) (J × L) R) (i : I) (j : J) (k : K) (l : L) : (comp I J K L R).symm n i j k l = n (i, k) (j, l)

@[simp]

theorem Matrix.comp_apply (I : Type u_1) (J : Type u_2) (K : Type u_3) (L : Type u_4) (R : Type u_5) (m : Matrix I J (Matrix K L R)) (ik : I × K) (jl : J × L) : (comp I J K L R) m ik jl = m ik.1 jl.1 ik.2 jl.2

theorem Matrix.comp_one {I : Type u_1} {J : Type u_2} {R : Type u_5} [DecidableEq I] [DecidableEq J] [Zero R] [One R] : (comp I I J J R) 1 = 1

theorem Matrix.comp_map_map {I : Type u_1} {J : Type u_2} {K : Type u_3} {L : Type u_4} {R : Type u_5} (R' : Type u_6) (M : Matrix I J (Matrix K L R)) (f : R → R') : (comp I J K L R') (M.map fun (M' : Matrix K L R) => M'.map f) = ((comp I J K L R) M).map f

@[simp]

theorem Matrix.comp_single_single {I : Type u_1} {J : Type u_2} {K : Type u_3} {L : Type u_4} {R : Type u_5} [DecidableEq I] [DecidableEq J] [DecidableEq K] [DecidableEq L] [Zero R] (i : I) (j : J) (k : K) (l : L) (r : R) : (comp I J K L R) (single i j (single k l r)) = single (i, k) (j, l) r

@[simp]

theorem Matrix.comp_symm_single {I : Type u_1} {J : Type u_2} {K : Type u_3} {L : Type u_4} {R : Type u_5} [DecidableEq I] [DecidableEq J] [DecidableEq K] [DecidableEq L] [Zero R] (ii : I × K) (jj : J × L) (r : R) : (comp I J K L R).symm (single ii jj r) = single ii.1 jj.1 (single ii.2 jj.2 r)

@[simp]

theorem Matrix.comp_diagonal_diagonal {I : Type u_1} {J : Type u_2} {R : Type u_5} [DecidableEq I] [DecidableEq J] [Zero R] (d : I → J → R) : (comp I I J J R) (diagonal fun (i : I) => diagonal fun (j : J) => d i j) = diagonal fun (ij : I × J) => d ij.1 ij.2

@[simp]

theorem Matrix.comp_symm_diagonal {I : Type u_1} {J : Type u_2} {R : Type u_5} [DecidableEq I] [DecidableEq J] [Zero R] (d : I × J → R) : (comp I I J J R).symm (diagonal d) = diagonal fun (i : I) => diagonal fun (j : J) => d (i, j)

theorem Matrix.comp_transpose {I : Type u_1} {J : Type u_2} {K : Type u_3} {L : Type u_4} {R : Type u_5} (M : Matrix I J (Matrix K L R)) : (comp J I K L R) M.transpose = ((comp I J L K R) (M.map fun (x : Matrix K L R) => x.transpose)).transpose

theorem Matrix.comp_map_transpose {I : Type u_1} {J : Type u_2} {K : Type u_3} {L : Type u_4} {R : Type u_5} (M : Matrix I J (Matrix K L R)) : (comp I J L K R) (M.map fun (x : Matrix K L R) => x.transpose) = ((comp J I K L R) M.transpose).transpose

theorem Matrix.comp_symm_transpose {I : Type u_1} {J : Type u_2} {K : Type u_3} {L : Type u_4} {R : Type u_5} (M : Matrix (I × K) (J × L) R) : (comp J I L K R).symm M.transpose = (((comp I J K L R).symm M).map fun (x : Matrix K L R) => x.transpose).transpose

theorem Matrix.transpose_comp {I : Type u_1} {J : Type u_2} {K : Type u_3} {L : Type u_4} {R : Type u_5} (M : Matrix I J (Matrix K L R)) : ((comp I J K L R) M).transpose = (comp J I L K R) (M.transpose.map fun (x : Matrix K L R) => x.transpose)

def Matrix.compAddEquiv (I : Type u_1) (J : Type u_2) (K : Type u_3) (L : Type u_4) (R : Type u_5) [Add R] : Matrix I J (Matrix K L R) ≃+ Matrix (I × K) (J × L) R

Matrix.comp as AddEquiv

Equations

• Matrix.compAddEquiv I J K L R = { toEquiv := Matrix.comp I J K L R, map_add' := ⋯ }

@[simp]

theorem Matrix.compAddEquiv_apply (I : Type u_1) (J : Type u_2) (K : Type u_3) (L : Type u_4) (R : Type u_5) [Add R] (M : Matrix I J (Matrix K L R)) : (compAddEquiv I J K L R) M = (comp I J K L R) M

@[simp]

theorem Matrix.compAddEquiv_symm_apply (I : Type u_1) (J : Type u_2) (K : Type u_3) (L : Type u_4) (R : Type u_5) [Add R] (M : Matrix (I × K) (J × L) R) : (compAddEquiv I J K L R).symm M = (comp I J K L R).symm M

def Matrix.compRingEquiv (I : Type u_1) (J : Type u_2) (R : Type u_5) [AddCommMonoid R] [Mul R] [Fintype I] [Fintype J] : Matrix I I (Matrix J J R) ≃+* Matrix (I × J) (I × J) R

Matrix.comp as RingEquiv

Equations

• Matrix.compRingEquiv I J R = { toEquiv := (Matrix.compAddEquiv I I J J R).toEquiv, map_mul' := ⋯, map_add' := ⋯ }

@[simp]

theorem Matrix.compRingEquiv_apply (I : Type u_1) (J : Type u_2) (R : Type u_5) [AddCommMonoid R] [Mul R] [Fintype I] [Fintype J] (M : Matrix I I (Matrix J J R)) : (compRingEquiv I J R) M = (comp I I J J R) M

@[simp]

theorem Matrix.compRingEquiv_symm_apply (I : Type u_1) (J : Type u_2) (R : Type u_5) [AddCommMonoid R] [Mul R] [Fintype I] [Fintype J] (M : Matrix (I × J) (I × J) R) : (compRingEquiv I J R).symm M = (comp I I J J R).symm M

instance Matrix.instIsStablyFiniteRing (I : Type u_1) [Fintype I] (R : Type u_7) [MulOne R] [AddCommMonoid R] [DecidableEq I] [IsStablyFiniteRing R] : IsStablyFiniteRing (Matrix I I R)

def Matrix.compLinearEquiv (I : Type u_1) (J : Type u_2) (K : Type u_3) (L : Type u_4) (R : Type u_5) (R₀ : Type u_7) [Semiring R₀] [AddCommMonoid R] [Module R₀ R] : Matrix I J (Matrix K L R) ≃ₗ[R₀] Matrix (I × K) (J × L) R

Matrix.comp as LinearEquiv

Equations

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

@[simp]

theorem Matrix.compLinearEquiv_apply (I : Type u_1) (J : Type u_2) (K : Type u_3) (L : Type u_4) (R : Type u_5) (R₀ : Type u_7) [Semiring R₀] [AddCommMonoid R] [Module R₀ R] (a✝ : Matrix I J (Matrix K L R)) : (compLinearEquiv I J K L R R₀) a✝ = (comp I J K L R) a✝

@[simp]

theorem Matrix.compLinearEquiv_symm_apply (I : Type u_1) (J : Type u_2) (K : Type u_3) (L : Type u_4) (R : Type u_5) (R₀ : Type u_7) [Semiring R₀] [AddCommMonoid R] [Module R₀ R] (a✝ : Matrix (I × K) (J × L) R) : (compLinearEquiv I J K L R R₀).symm a✝ = (comp I J K L R).symm a✝

def Matrix.compAlgEquiv (I : Type u_1) (J : Type u_2) (R : Type u_5) (K : Type u_7) [CommSemiring K] [Semiring R] [Fintype I] [Fintype J] [Algebra K R] [DecidableEq I] [DecidableEq J] : Matrix I I (Matrix J J R) ≃ₐ[K] Matrix (I × J) (I × J) R

Matrix.comp as AlgEquiv

Equations

• Matrix.compAlgEquiv I J R K = { toEquiv := (Matrix.compRingEquiv I J R).toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem Matrix.compAlgEquiv_apply (I : Type u_1) (J : Type u_2) (R : Type u_5) (K : Type u_7) [CommSemiring K] [Semiring R] [Fintype I] [Fintype J] [Algebra K R] [DecidableEq I] [DecidableEq J] (M : Matrix I I (Matrix J J R)) : (compAlgEquiv I J R K) M = (comp I I J J R) M

@[simp]

theorem Matrix.compAlgEquiv_symm_apply (I : Type u_1) (J : Type u_2) (R : Type u_5) (K : Type u_7) [CommSemiring K] [Semiring R] [Fintype I] [Fintype J] [Algebra K R] [DecidableEq I] [DecidableEq J] (M : Matrix (I × J) (I × J) R) : (compAlgEquiv I J R K).symm M = (comp I I J J R).symm M

@[simp]

theorem Matrix.isUnit_comp_iff (I : Type u_1) (J : Type u_2) (R : Type u_5) [Semiring R] [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] {M : Matrix I I (Matrix J J R)} : IsUnit ((comp I I J J R) M) ↔ IsUnit M

@[simp]

theorem Matrix.isUnit_comp_symm_iff (I : Type u_1) (J : Type u_2) (R : Type u_5) [Semiring R] [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] {M : Matrix (I × J) (I × J) R} : IsUnit ((comp I I J J R).symm M) ↔ IsUnit M