det_to_matrix_orthonormalBasis — Mathlib

English

Inverse of equivalence corresponds to inverse toBasis mapping.

Русский

Обратная эквивалентность соответствует обратному отображению toBasis.

LaTeX

$$$$ (b.equiv b' e)^{-1}.toBasis = (b'.equiv b e^{-1}).toBasis. $$$$

Lean4

/-- The determinant of the change-of-basis matrix between two orthonormal bases `a`, `b` has
unit length. -/
@[simp]
theorem det_to_matrix_orthonormalBasis : ‖a.toBasis.det b‖ = 1 :=
  by
  have := (Matrix.det_of_mem_unitary (a.toMatrix_orthonormalBasis_mem_unitary b)).2
  rw [star_def, RCLike.mul_conj] at this
  norm_cast at this
  rwa [pow_eq_one_iff_of_nonneg (norm_nonneg _) two_ne_zero] at this

Похожие утверждения