Definition:Non-Negative Definite Mapping
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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
Also see
- Definition:Semi-Inner Product, where this property is used in the definition of the concept.
Linguistic Note
The property, as a noun, of a Non-Negative Definite Mapping, is referred to as non-negative definiteness.
Sources
- 1989: R.M. Dudley: Real Analysis and Probability: $\S 5.3$: Hilbert Spaces