English
All nondegenerate quadratic forms over the complex field are equivalent to each other; more precisely, if Q1 and Q2 are nondegenerate quadratic forms on a complex vector space, there exists an equivalence between them.
Русский
Все неdegenerate квадратичные формы над комплексными числами эквивалентны друг другу; существует квадратичный эквивалент между любыми двумя формами.
LaTeX
$$$ Q_1 \\sim Q_2 $ for nondegenerate complex quadratic forms.$$
Lean4
/-- All nondegenerate quadratic forms on the complex numbers are equivalent. -/
theorem complex_equivalent {M : Type*} [AddCommGroup M] [Module ℂ M] [FiniteDimensional ℂ M] (Q₁ Q₂ : QuadraticForm ℂ M)
(hQ₁ : (associated (R := ℂ) Q₁).SeparatingLeft) (hQ₂ : (associated (R := ℂ) Q₂).SeparatingLeft) :
Equivalent Q₁ Q₂ :=
(Q₁.equivalent_sum_squares hQ₁).trans (Q₂.equivalent_sum_squares hQ₂).symm
