isPosSemidef_iff — Mathlib

English

Let B be a bilinear form on an R-module M. Then B is positive semidefinite if and only if it is positive semidefinite when viewed as a sesquilinear form.

Русский

Пусть B — билинейная форма на модуле M над кольцом R. Тогда B положительно полуопределена тогда и только тогда, когда она так же определена как сескалинейная форма.

LaTeX

$$$B.IsPosSemidef \\iff LinearMap.IsPosSemidef B$$$

Lean4

/-- A bilinear form is positive semidefinite if and only if it is positive semidefinite
  as a sesquilinear form. -/
theorem isPosSemidef_iff [LE R] {B : BilinForm R M} : B.IsPosSemidef ↔ LinearMap.IsPosSemidef B :=
  isPosSemidef_def.trans <| (isSymm_iff.and isNonneg_iff).trans LinearMap.isPosSemidef_def.symm

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