English
Each eigenvector v_i of A with eigenvalue λ_i satisfies A v_i = λ_i v_i.
Русский
Каждый собственный вектор v_i эрмитовой матрицы удовлетворяет A v_i = λ_i v_i.
LaTeX
$$$A \cdot v_i = \lambda_i v_i$ for each eigenvector $v_i$ in the eigenvectorBasis.$$
Lean4
theorem mulVec_eigenvectorBasis (j : n) :
A *ᵥ ⇑(hA.eigenvectorBasis j) = (hA.eigenvalues j) • ⇑(hA.eigenvectorBasis j) := by
simpa only [eigenvectorBasis, OrthonormalBasis.reindex_apply, toEuclideanLin_apply,
RCLike.real_smul_eq_coe_smul (K := 𝕜)] using
congr(⇑$((isHermitian_iff_isSymmetric.1 hA).apply_eigenvectorBasis finrank_euclideanSpace
((Fintype.equivOfCardEq (Fintype.card_fin _)).symm j)))