Definition:Non-Negative Definite Mapping

Definition
Let $\C$ be the field of complex numbers.

Let $\F$ be a subfield of $\C$.

Let $V$ be a vector space over $\F$

Let $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ be a mapping.

Then $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ is non-negative definite iff:


 * $\forall x \in V: \quad \left \langle {x, x} \right \rangle \in \R_{\geq 0}$

That is, the image of $\left \langle {x, x} \right \rangle$ is always a non-negative real number.

Also known as

 * Nonnegative definite mapping

Also see

 * Definition:Semi-Inner Product, where this property is used in the definition of the concept.

Linguistic Note
This property, as a noun, is referred to as non-negative definiteness.