transpose_pow — Mathlib

English

Transpose commutes with powers: (M^k)^T = (M^T)^k for square matrices over a commutative semiring.

Русский

Транспонирование commute с степенями: (M^k)^T = (M^T)^k для квадратных матриц над коммутативной полусходной.

LaTeX

$$$\left(M^k\right)^T = \left(M^T\right)^k$ for all k ∈ ℕ and M ∈ M_{m\times m}(\alpha).$$

Lean4

@[simp]
theorem transpose_pow [CommSemiring α] [Fintype m] [DecidableEq m] (M : Matrix m m α) (k : ℕ) : (M ^ k)ᵀ = Mᵀ ^ k :=
  MulOpposite.op_injective <| map_pow (transposeRingEquiv m α) M k

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