Positive Definiteness

linear-algebra

Definition

Positive Definiteness

A map $⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{F}$ on a vector space $V$ over $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$ is positive definite if

$$⟨v,v⟩\ge 0\forall v\in V,$$

with equality iff $v$ is the zero vector:

$$⟨v,v⟩=0⟺v=0.$$