Theorems · Theorem · operator theory

LinearMap.posSemidef_toMatrix_iff

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {ι : Type u_4} [inst_3 : Fintype ι] [inst_4 : DecidableEq ι] {A : E →ₗ[𝕜] E} (b : OrthonormalBasis ι 𝕜 E),
  ((LinearMap.toMatrix b.toBasis b.toBasis) A).PosSemidef ↔ A.IsPositive

A.toMatrix is positive semi-definite if and only if A is positive.

Defined in

Mathlib.Analysis.InnerProductSpace.Positive

Cited by

1 results in Mathlib

Foundations

Depth 236 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFintypeDecidableEq