partial equivalences for matrices

Using partial equivalences to represent matrices. This file introduces the function PEquiv.toMatrix, which returns a matrix containing ones and zeros. For any partial equivalence f, f.toMatrix i j = 1 ↔ f i = some j.

The following important properties of this function are proved toMatrix_trans : (f.trans g).toMatrix = f.toMatrix * g.toMatrix toMatrix_symm : f.symm.toMatrix = f.toMatrixᵀ toMatrix_refl : (PEquiv.refl n).toMatrix = 1 toMatrix_bot : ⊥.toMatrix = 0

This theory gives the matrix representation of projection linear maps, and their right inverses. For example, the matrix (single (0 : Fin 1) (i : Fin n)).toMatrix corresponds to the ith projection map from R^n to R.

Any injective function Fin m → Fin n gives rise to a PEquiv, whose matrix is the projection map from R^m → R^n represented by the same function. The transpose of this matrix is the right inverse of this map, sending anything not in the image to zero.

notations

This file uses ᵀ for Matrix.transpose.

def PEquiv.toMatrix {m : Type u_3} {n : Type u_4} {α : Type v} [DecidableEq n] [Zero α] [One α] (f : m ≃. n) : Matrix m n α

toMatrix returns a matrix containing ones and zeros. f.toMatrix i j is 1 if f i = some j and 0 otherwise

Equations

• f.toMatrix = Matrix.of fun (i : m) (j : n) => if j ∈ f i then 1 else 0

@[simp]

theorem PEquiv.toMatrix_apply {m : Type u_3} {n : Type u_4} {α : Type v} [DecidableEq n] [Zero α] [One α] (f : m ≃. n) (i : m) (j : n) : f.toMatrix i j = if j ∈ f i then 1 else 0

theorem PEquiv.mul_matrix_apply {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type v} [Fintype m] [DecidableEq m] [Semiring α] (f : l ≃. m) (M : Matrix m n α) (i : l) (j : n) : (f.toMatrix * M) i j = Option.casesOn (f i) 0 fun (fi : m) => M fi j

theorem PEquiv.toMatrix_symm {m : Type u_3} {n : Type u_4} {α : Type v} [DecidableEq m] [DecidableEq n] [Zero α] [One α] (f : m ≃. n) : f.symm.toMatrix = f.toMatrix.transpose

@[simp]

theorem PEquiv.toMatrix_refl {n : Type u_4} {α : Type v} [DecidableEq n] [Zero α] [One α] : (PEquiv.refl n).toMatrix = 1

theorem PEquiv.matrix_mul_apply {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type v} [Fintype m] [Semiring α] [DecidableEq n] (M : Matrix l m α) (f : m ≃. n) (i : l) (j : n) : (M * f.toMatrix) i j = Option.casesOn (f.symm j) 0 fun (fj : m) => M i fj

theorem PEquiv.toPEquiv_mul_matrix {m : Type u_3} {n : Type u_4} {α : Type v} [Fintype m] [DecidableEq m] [Semiring α] (f : m ≃ m) (M : Matrix m n α) : f.toPEquiv.toMatrix * M = M.submatrix (⇑f) id

theorem PEquiv.mul_toPEquiv_toMatrix {m : Type u_5} {n : Type u_6} {α : Type u_7} [Fintype n] [DecidableEq n] [Semiring α] (f : n ≃ n) (M : Matrix m n α) : M * f.toPEquiv.toMatrix = M.submatrix id ⇑f.symm

theorem PEquiv.toMatrix_trans {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type v} [Fintype m] [DecidableEq m] [DecidableEq n] [Semiring α] (f : l ≃. m) (g : m ≃. n) : (f.trans g).toMatrix = f.toMatrix * g.toMatrix

@[simp]

theorem PEquiv.toMatrix_bot {m : Type u_3} {n : Type u_4} {α : Type v} [DecidableEq n] [Zero α] [One α] : ⊥.toMatrix = 0

theorem PEquiv.toMatrix_injective {m : Type u_3} {n : Type u_4} {α : Type v} [DecidableEq n] [MonoidWithZero α] [Nontrivial α] : Function.Injective PEquiv.toMatrix

theorem PEquiv.toMatrix_swap {n : Type u_4} {α : Type v} [DecidableEq n] [Ring α] (i j : n) : (Equiv.toPEquiv (Equiv.swap i j)).toMatrix = 1 - (PEquiv.single i i).toMatrix - (PEquiv.single j j).toMatrix + (PEquiv.single i j).toMatrix + (PEquiv.single j i).toMatrix

@[simp]

theorem PEquiv.single_mul_single {k : Type u_1} {m : Type u_3} {n : Type u_4} {α : Type v} [Fintype n] [DecidableEq k] [DecidableEq m] [DecidableEq n] [Semiring α] (a : m) (b : n) (c : k) : (PEquiv.single a b).toMatrix * (PEquiv.single b c).toMatrix = (PEquiv.single a c).toMatrix

theorem PEquiv.single_mul_single_of_ne {k : Type u_1} {m : Type u_3} {n : Type u_4} {α : Type v} [Fintype n] [DecidableEq n] [DecidableEq k] [DecidableEq m] [Semiring α] {b₁ b₂ : n} (hb : b₁ ≠ b₂) (a : m) (c : k) : (PEquiv.single a b₁).toMatrix * (PEquiv.single b₂ c).toMatrix = 0

@[simp]

theorem PEquiv.single_mul_single_right {k : Type u_1} {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type v} [Fintype n] [Fintype k] [DecidableEq n] [DecidableEq k] [DecidableEq m] [Semiring α] (a : m) (b : n) (c : k) (M : Matrix k l α) : (PEquiv.single a b).toMatrix * ((PEquiv.single b c).toMatrix * M) = (PEquiv.single a c).toMatrix * M

Restatement of single_mul_single, which will simplify expressions in simp normal form, when associativity may otherwise need to be carefully applied.

theorem PEquiv.equiv_toPEquiv_toMatrix {n : Type u_4} {α : Type v} [DecidableEq n] [Zero α] [One α] (σ : n ≃ n) (i j : n) : σ.toPEquiv.toMatrix i j = 1 (σ i) j

We can also define permutation matrices by permuting the rows of the identity matrix.