Matrices

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def DMatrix (m : Type u) (n : Type u') [inst : Fintype m] [inst : Fintype n] (α : m → n → Type v) : Type (maxuu'v)

DMatrix m n is the type of dependently typed matrices whose rows are indexed by the fintype m and whose columns are indexed by the fintype n.

Equations

• DMatrix m n α = ((i : m) → (j : n) → α i j)

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theorem DMatrix.ext_iff {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} {M : DMatrix m n α} {N : DMatrix m n α} : (∀ (i : m) (j : n), M i j = N i j) ↔ M = N

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theorem DMatrix.ext {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} {M : DMatrix m n α} {N : DMatrix m n α} : (∀ (i : m) (j : n), M i j = N i j) → M = N

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def DMatrix.map {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} (M : DMatrix m n α) {β : m → n → Type w} (f : ⦃i : m⦄ → ⦃j : n⦄ → α i j → β i j) : DMatrix m n β

M.map f is the DMatrix obtained by applying f to each entry of the matrix M.

Equations

• DMatrix.map M f i j = f i j (M i j)

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theorem DMatrix.map_apply {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} {M : DMatrix m n α} {β : m → n → Type w} {f : ⦃i : m⦄ → ⦃j : n⦄ → α i j → β i j} {i : m} {j : n} : DMatrix.map M f i j = f i j (M i j)

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@[simp]

theorem DMatrix.map_map {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} {M : DMatrix m n α} {β : m → n → Type w} {γ : m → n → Type z} {f : ⦃i : m⦄ → ⦃j : n⦄ → α i j → β i j} {g : ⦃i : m⦄ → ⦃j : n⦄ → β i j → γ i j} : DMatrix.map (DMatrix.map M f) g = DMatrix.map M fun i j x => g i j (f i j x)

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def DMatrix.transpose {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} (M : DMatrix m n α) : DMatrix n m fun j i => α i j

The transpose of a dmatrix.

Equations

• DMatrix.transpose M x x = match x, x with | x, y => M y x

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def DMatrix.«term_ᵀ» : Lean.TrailingParserDescr

The transpose of a dmatrix.

Equations

• DMatrix.«term_ᵀ» = Lean.ParserDescr.trailingNode `DMatrix.term_ᵀ 1024 1024 (Lean.ParserDescr.symbol "ᵀ")

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def DMatrix.col {m : Type u_1} [inst : Fintype m] {α : m → Type v} (w : (i : m) → α i) : DMatrix m Unit fun i _j => α i

dmatrix.col u is the column matrix whose entries are given by u.

Equations

• DMatrix.col w x x = match x, x with | x, _y => w x

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def DMatrix.row {n : Type u_1} [inst : Fintype n] {α : n → Type v} (v : (j : n) → α j) : DMatrix Unit n fun _i j => α j

dmatrix.row u is the row matrix whose entries are given by u.

Equations

• DMatrix.row v x x = match x, x with | _x, y => v y

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instance DMatrix.instInhabitedDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → Inhabited (α i j)] : Inhabited (DMatrix m n α)

Equations

• DMatrix.instInhabitedDMatrix = { default := fun i j => default }

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instance DMatrix.instAddDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → Add (α i j)] : Add (DMatrix m n α)

Equations

• DMatrix.instAddDMatrix = Pi.instAdd

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instance DMatrix.instAddSemigroupDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → AddSemigroup (α i j)] : AddSemigroup (DMatrix m n α)

Equations

• DMatrix.instAddSemigroupDMatrix = Pi.addSemigroup

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instance DMatrix.instAddCommSemigroupDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → AddCommSemigroup (α i j)] : AddCommSemigroup (DMatrix m n α)

Equations

• DMatrix.instAddCommSemigroupDMatrix = Pi.addCommSemigroup

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instance DMatrix.instZeroDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → Zero (α i j)] : Zero (DMatrix m n α)

Equations

• DMatrix.instZeroDMatrix = Pi.instZero

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instance DMatrix.instAddMonoidDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → AddMonoid (α i j)] : AddMonoid (DMatrix m n α)

Equations

• DMatrix.instAddMonoidDMatrix = Pi.addMonoid

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instance DMatrix.instAddCommMonoidDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → AddCommMonoid (α i j)] : AddCommMonoid (DMatrix m n α)

Equations

• DMatrix.instAddCommMonoidDMatrix = Pi.addCommMonoid

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instance DMatrix.instNegDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → Neg (α i j)] : Neg (DMatrix m n α)

Equations

• DMatrix.instNegDMatrix = Pi.instNeg

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instance DMatrix.instSubDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → Sub (α i j)] : Sub (DMatrix m n α)

Equations

• DMatrix.instSubDMatrix = Pi.instSub

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instance DMatrix.instAddGroupDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → AddGroup (α i j)] : AddGroup (DMatrix m n α)

Equations

• DMatrix.instAddGroupDMatrix = Pi.addGroup

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instance DMatrix.instAddCommGroupDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → AddCommGroup (α i j)] : AddCommGroup (DMatrix m n α)

Equations

• DMatrix.instAddCommGroupDMatrix = Pi.addCommGroup

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instance DMatrix.instUniqueDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → Unique (α i j)] : Unique (DMatrix m n α)

Equations

• DMatrix.instUniqueDMatrix = Pi.unique

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instance DMatrix.instSubsingletonDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : ∀ (i : m) (j : n), Subsingleton (α i j)] : Subsingleton (DMatrix m n α)

Equations

• @DMatrix.instSubsingletonDMatrix = @DMatrix.instSubsingletonDMatrix.proof_1

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theorem DMatrix.zero_apply {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → Zero (α i j)] (i : m) (j : n) : OfNat.ofNat (DMatrix m n α) 0 Zero.toOfNat0 i j = 0

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theorem DMatrix.neg_apply {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → Neg (α i j)] (M : DMatrix m n α) (i : m) (j : n) : (-DMatrix m n α) DMatrix.instNegDMatrix M i j = -M i j

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theorem DMatrix.add_apply {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → Add (α i j)] (M : DMatrix m n α) (N : DMatrix m n α) (i : m) (j : n) : (DMatrix m n α + DMatrix m n α) (DMatrix m n α) instHAdd M N i j = M i j + N i j

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theorem DMatrix.sub_apply {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → Sub (α i j)] (M : DMatrix m n α) (N : DMatrix m n α) (i : m) (j : n) : (DMatrix m n α - DMatrix m n α) (DMatrix m n α) instHSub M N i j = M i j - N i j

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theorem DMatrix.map_zero {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → Zero (α i j)] {β : m → n → Type w} [inst : (i : m) → (j : n) → Zero (β i j)] {f : ⦃i : m⦄ → ⦃j : n⦄ → α i j → β i j} (h : ∀ (i : m) (j : n), f i j 0 = 0) : DMatrix.map 0 f = 0

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theorem DMatrix.map_add {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → AddMonoid (α i j)] {β : m → n → Type w} [inst : (i : m) → (j : n) → AddMonoid (β i j)] (f : ⦃i : m⦄ → ⦃j : n⦄ → α i j →+ β i j) (M : DMatrix m n α) (N : DMatrix m n α) : (DMatrix.map (M + N) fun i j => ↑(f i j)) = (DMatrix.map M fun i j => ↑(f i j)) + DMatrix.map N fun i j => ↑(f i j)

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theorem DMatrix.map_sub {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → AddGroup (α i j)] {β : m → n → Type w} [inst : (i : m) → (j : n) → AddGroup (β i j)] (f : ⦃i : m⦄ → ⦃j : n⦄ → α i j →+ β i j) (M : DMatrix m n α) (N : DMatrix m n α) : (DMatrix.map (M - N) fun i j => ↑(f i j)) = (DMatrix.map M fun i j => ↑(f i j)) - DMatrix.map N fun i j => ↑(f i j)

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instance DMatrix.subsingleton_of_empty_left {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : IsEmpty m] : Subsingleton (DMatrix m n α)

Equations

• @DMatrix.subsingleton_of_empty_left = @DMatrix.subsingleton_of_empty_left.proof_1

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instance DMatrix.subsingleton_of_empty_right {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : IsEmpty n] : Subsingleton (DMatrix m n α)

Equations

• @DMatrix.subsingleton_of_empty_right = @DMatrix.subsingleton_of_empty_right.proof_1

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def AddMonoidHom.mapDMatrix {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → AddMonoid (α i j)] {β : m → n → Type w} [inst : (i : m) → (j : n) → AddMonoid (β i j)] (f : ⦃i : m⦄ → ⦃j : n⦄ → α i j →+ β i j) : DMatrix m n α →+ DMatrix m n β

The AddMonoidHom between spaces of dependently typed matrices induced by an AddMonoidHom between their coefficients.

Equations

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

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theorem AddMonoidHom.mapDMatrix_apply {m : Type u_1} {n : Type u_2} [inst : Fintype m] [inst : Fintype n] {α : m → n → Type v} [inst : (i : m) → (j : n) → AddMonoid (α i j)] {β : m → n → Type w} [inst : (i : m) → (j : n) → AddMonoid (β i j)] (f : ⦃i : m⦄ → ⦃j : n⦄ → α i j →+ β i j) (M : DMatrix m n α) : ↑(AddMonoidHom.mapDMatrix f) M = DMatrix.map M fun i j => ↑(f i j)