toMatrix_orthonormalBasis_self_mul_conjTranspose

English

Equiv in Euclidean space expresses the same sum-of-coordinates expansion using basisFun.

Русский

Эквивалентность Евклидова пространства выражает тот же разложение по координатам через basisFun.

LaTeX

$$$$ \\mathrm{equiv\\_euclidean}(x) = \\sum_i x_i b_i. $$$$

Lean4

/-- A version of `OrthonormalBasis.toMatrix_orthonormalBasis_mem_unitary` that works for bases with
different index types. -/
@[simp]
theorem toMatrix_orthonormalBasis_self_mul_conjTranspose [Fintype ι'] (a : OrthonormalBasis ι 𝕜 E)
    (b : OrthonormalBasis ι' 𝕜 E) : a.toBasis.toMatrix b * (a.toBasis.toMatrix b)ᴴ = 1 := by
  classical
    rw [Matrix.mul_eq_one_comm_of_equiv (a.toBasis.indexEquiv b.toBasis),
    a.toMatrix_orthonormalBasis_conjTranspose_mul_self b]

Похожие утверждения