# 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 $\innerprod \cdot \cdot: V \times V \to \mathbb F$ be a mapping.

Then $\innerprod \cdot \cdot: V \times V \to \mathbb F$ is non-negative definite if and only if:

$\forall x \in V: \innerprod x x \in \R_{\ge 0}$

That is, the image of $\innerprod x x$ is always a non-negative real number.

## Also known as

• Nonnegative definite mapping

## Linguistic Note

The property, as a noun, of a Non-Negative Definite Mapping, is referred to as non-negative definiteness.