Real quadratic forms

Sylvester's law of inertia equivalent_one_neg_one_weighted_sum_squared: A real quadratic form is equivalent to a weighted sum of squares with the weights being ±1 or 0.

When the real quadratic form is nondegenerate we can take the weights to be ±1, as in equivalent_one_zero_neg_one_weighted_sum_squared.

noncomputable def QuadraticForm.isometryEquivSignWeightedSumSquares {ι : Type u_1} [Fintype ι] [DecidableEq ι] (w : ι → ℝ) : QuadraticForm.IsometryEquiv (QuadraticForm.weightedSumSquares ℝ w) (QuadraticForm.weightedSumSquares ℝ (Real.sign ∘ w))

The isometry between a weighted sum of squares with weights u on the (non-zero) real numbers and the weighted sum of squares with weights sign ∘ u.

Equations

• One or more equations did not get rendered due to their size.

theorem QuadraticForm.equivalent_one_neg_one_weighted_sum_squared {M : Type u_2} [AddCommGroup M] [Module ℝ M] [FiniteDimensional ℝ M] (Q : QuadraticForm ℝ M) (hQ : BilinForm.Nondegenerate (QuadraticForm.associated Q)) : ∃ (w : Fin (FiniteDimensional.finrank ℝ M) → ℝ), (∀ (i : Fin (FiniteDimensional.finrank ℝ M)), w i = -1 ∨ w i = 1) ∧ QuadraticForm.Equivalent Q (QuadraticForm.weightedSumSquares ℝ w)

Sylvester's law of inertia: A nondegenerate real quadratic form is equivalent to a weighted sum of squares with the weights being ±1.

theorem QuadraticForm.equivalent_one_zero_neg_one_weighted_sum_squared {M : Type u_2} [AddCommGroup M] [Module ℝ M] [FiniteDimensional ℝ M] (Q : QuadraticForm ℝ M) : ∃ (w : Fin (FiniteDimensional.finrank ℝ M) → ℝ), (∀ (i : Fin (FiniteDimensional.finrank ℝ M)), w i = -1 ∨ w i = 0 ∨ w i = 1) ∧ QuadraticForm.Equivalent Q (QuadraticForm.weightedSumSquares ℝ w)

Sylvester's law of inertia: A real quadratic form is equivalent to a weighted sum of squares with the weights being ±1 or 0.