polarBilin_prod — Mathlib

English

The bilinear form (polarBil in) of the product is the sum of the bilinear forms applied to the left and right projections.

Русский

bilirropic двукратная форма произведения равна сумме bilinear форм, применённых к левому и правому проекциям.

LaTeX

$$$ (Q_1 \\text{ prod } Q_2).\\mathrm{polarBilin} = (Q_1.\\mathrm{polarBilin})\\circ (\\mathrm{fst},\\mathrm{fst}) + (Q_2.\\mathrm{polarBilin})\\circ (\\mathrm{snd},\\mathrm{snd}). $$$

Lean4

@[simp]
theorem polarBilin_prod (Q₁ : QuadraticMap R M₁ P) (Q₂ : QuadraticMap R M₂ P) :
    (Q₁.prod Q₂).polarBilin =
      Q₁.polarBilin.compl₁₂ (.fst R M₁ M₂) (.fst R M₁ M₂) + Q₂.polarBilin.compl₁₂ (.snd R M₁ M₂) (.snd R M₁ M₂) :=
  LinearMap.ext₂ <| polar_prod _ _

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