Theorems · Theorem · linear algebra

Matrix.posDef_gram_of_linearIndependent

∀ {E : Type u_1} {n : Type u_2} {𝕜 : Type u_4} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace 𝕜 E] [Finite n] {v : n → E}, LinearIndependent 𝕜 v → (Matrix.gram 𝕜 v).PosDef

In a normed space, linear independence of v implies positive definiteness of gram 𝕜 v.

Defined in

Mathlib.Analysis.InnerProductSpace.GramMatrix

Cited by

1 results in Mathlib

Foundations

Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFinite