Circulant matrices

This file contains the definition and basic results about circulant matrices. Given a vector v : n → α indexed by a type that is endowed with subtraction, Matrix.circulant v is the matrix whose (i, j)th entry is v (i - j).

Main results

• Matrix.circulant: the circulant matrix generated by a given vector v : n → α.
• Matrix.circulant_mul: the product of two circulant matrices circulant v and circulant w is the circulant matrix generated by circulant v *ᵥ w.
• Matrix.circulant_mul_comm: multiplication of circulant matrices commutes when the elements do.

Implementation notes

Matrix.Fin.foo is the Fin n version of Matrix.foo. Namely, the index type of the circulant matrices in discussion is Fin n.

Tags

circulant, matrix

def Matrix.circulant {α : Type u_1} {n : Type u_4} [Sub n] (v : n → α) : Matrix n n α

Given the condition [Sub n] and a vector v : n → α, we define circulant v to be the circulant matrix generated by v of type Matrix n n α. The (i,j)th entry is defined to be v (i - j).

Equations

• Matrix.circulant v = Matrix.of fun (i j : n) => v (i - j)

@[simp]

theorem Matrix.circulant_apply {α : Type u_1} {n : Type u_4} [Sub n] (v : n → α) (i : n) (j : n) : Matrix.circulant v i j = v (i - j)

theorem Matrix.circulant_col_zero_eq {α : Type u_1} {n : Type u_4} [AddGroup n] (v : n → α) (i : n) : Matrix.circulant v i 0 = v i

theorem Matrix.circulant_injective {α : Type u_1} {n : Type u_4} [AddGroup n] : Function.Injective Matrix.circulant

theorem Matrix.Fin.circulant_injective {α : Type u_1} (n : ℕ) : Function.Injective fun (v : Fin n → α) => Matrix.circulant v

@[simp]

theorem Matrix.circulant_inj {α : Type u_1} {n : Type u_4} [AddGroup n] {v : n → α} {w : n → α} : Matrix.circulant v = Matrix.circulant w ↔ v = w

@[simp]

theorem Matrix.Fin.circulant_inj {α : Type u_1} {n : ℕ} {v : Fin n → α} {w : Fin n → α} : Matrix.circulant v = Matrix.circulant w ↔ v = w

theorem Matrix.transpose_circulant {α : Type u_1} {n : Type u_4} [AddGroup n] (v : n → α) : Matrix.transpose (Matrix.circulant v) = Matrix.circulant fun (i : n) => v (-i)

theorem Matrix.conjTranspose_circulant {α : Type u_1} {n : Type u_4} [Star α] [AddGroup n] (v : n → α) : Matrix.conjTranspose (Matrix.circulant v) = Matrix.circulant (star fun (i : n) => v (-i))

theorem Matrix.Fin.transpose_circulant {α : Type u_1} {n : ℕ} (v : Fin n → α) : Matrix.transpose (Matrix.circulant v) = Matrix.circulant fun (i : Fin n) => v (-i)

theorem Matrix.Fin.conjTranspose_circulant {α : Type u_1} [Star α] {n : ℕ} (v : Fin n → α) : Matrix.conjTranspose (Matrix.circulant v) = Matrix.circulant (star fun (i : Fin n) => v (-i))

theorem Matrix.map_circulant {α : Type u_1} {β : Type u_2} {n : Type u_4} [Sub n] (v : n → α) (f : α → β) : Matrix.map (Matrix.circulant v) f = Matrix.circulant fun (i : n) => f (v i)

theorem Matrix.circulant_neg {α : Type u_1} {n : Type u_4} [Neg α] [Sub n] (v : n → α) : Matrix.circulant (-v) = -Matrix.circulant v

@[simp]

theorem Matrix.circulant_zero (α : Type u_6) (n : Type u_7) [Zero α] [Sub n] : Matrix.circulant 0 = 0

theorem Matrix.circulant_add {α : Type u_1} {n : Type u_4} [Add α] [Sub n] (v : n → α) (w : n → α) : Matrix.circulant (v + w) = Matrix.circulant v + Matrix.circulant w

theorem Matrix.circulant_sub {α : Type u_1} {n : Type u_4} [Sub α] [Sub n] (v : n → α) (w : n → α) : Matrix.circulant (v - w) = Matrix.circulant v - Matrix.circulant w

theorem Matrix.circulant_mul {α : Type u_1} {n : Type u_4} [Semiring α] [Fintype n] [AddGroup n] (v : n → α) (w : n → α) : Matrix.circulant v * Matrix.circulant w = Matrix.circulant (Matrix.mulVec (Matrix.circulant v) w)

The product of two circulant matrices circulant v and circulant w is the circulant matrix generated by circulant v *ᵥ w.

theorem Matrix.Fin.circulant_mul {α : Type u_1} [Semiring α] {n : ℕ} (v : Fin n → α) (w : Fin n → α) : Matrix.circulant v * Matrix.circulant w = Matrix.circulant (Matrix.mulVec (Matrix.circulant v) w)

theorem Matrix.circulant_mul_comm {α : Type u_1} {n : Type u_4} [CommSemigroup α] [AddCommMonoid α] [Fintype n] [AddCommGroup n] (v : n → α) (w : n → α) : Matrix.circulant v * Matrix.circulant w = Matrix.circulant w * Matrix.circulant v

Multiplication of circulant matrices commutes when the elements do.

theorem Matrix.Fin.circulant_mul_comm {α : Type u_1} [CommSemigroup α] [AddCommMonoid α] {n : ℕ} (v : Fin n → α) (w : Fin n → α) : Matrix.circulant v * Matrix.circulant w = Matrix.circulant w * Matrix.circulant v

theorem Matrix.circulant_smul {α : Type u_1} {n : Type u_4} {R : Type u_5} [Sub n] [SMul R α] (k : R) (v : n → α) : Matrix.circulant (k • v) = k • Matrix.circulant v

k • circulant v is another circulant matrix circulant (k • v).

@[simp]

theorem Matrix.circulant_single_one (α : Type u_6) (n : Type u_7) [Zero α] [One α] [DecidableEq n] [AddGroup n] : Matrix.circulant (Pi.single 0 1) = 1

@[simp]

theorem Matrix.circulant_single {α : Type u_1} (n : Type u_6) [Semiring α] [DecidableEq n] [AddGroup n] [Fintype n] (a : α) : Matrix.circulant (Pi.single 0 a) = (Matrix.scalar n) a

theorem Matrix.Fin.circulant_ite (α : Type u_6) [Zero α] [One α] (n : ℕ) : (Matrix.circulant fun (i : Fin n) => if ↑i = 0 then 1 else 0) = 1

Note we use ↑i = 0 instead of i = 0 as Fin 0 has no 0. This means that we cannot state this with Pi.single as we did with Matrix.circulant_single.

theorem Matrix.circulant_isSymm_iff {α : Type u_1} {n : Type u_4} [AddGroup n] {v : n → α} : Matrix.IsSymm (Matrix.circulant v) ↔ ∀ (i : n), v (-i) = v i

A circulant of v is symmetric iff v equals its reverse.

theorem Matrix.Fin.circulant_isSymm_iff {α : Type u_1} {n : ℕ} {v : Fin n → α} : Matrix.IsSymm (Matrix.circulant v) ↔ ∀ (i : Fin n), v (-i) = v i

theorem Matrix.circulant_isSymm_apply {α : Type u_1} {n : Type u_4} [AddGroup n] {v : n → α} (h : Matrix.IsSymm (Matrix.circulant v)) (i : n) : v (-i) = v i

If circulant v is symmetric, ∀ i j : I, v (- i) = v i.

theorem Matrix.Fin.circulant_isSymm_apply {α : Type u_1} {n : ℕ} {v : Fin n → α} (h : Matrix.IsSymm (Matrix.circulant v)) (i : Fin n) : v (-i) = v i