star_vecMul — Mathlib

English

The star dot product identities give relationships between star of vectors and multiplication by matrices; star_mulVec expresses Aᴴ ·ᵥ x = star (star x ᵥ* A).

Русский

Связь между звездой и умножением вектора на матрицу: Aᴴ ·ᵥ x = star (star x ᵥ* A).

LaTeX

$$Theorem: circulant star-dot product identity for Fin-indexed vectors: (Aᴴ) ⋅ x = star (star x ⋅ A).$$

Lean4

theorem star_vecMul [Fintype m] [StarRing α] (M : Matrix m n α) (v : m → α) : star (v ᵥ* M) = Mᴴ *ᵥ star v :=
  funext fun _ => (star_dotProduct_star _ _).symm

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