Linear and multilinear algebra; matrix theory Definition

Positive definite quadratic form

Definition

Let $V$ be a finite-dimensional real vector space, and let $Q$ be a quadratic form on $V$ . The form $Q$ is positive definite when $Q(v) > 0 \qquad \text{for all } v \in V \text{ with } v \ne 0.$

Remarks

$Q(x) = x^T A x$ with $A$ symmetric is positive definite if and only if $A$ is a positive definite matrix. Equivalently, the signature of $Q$ is $(n, 0, 0)$ , with $n = \dim V$ . The form $x^2 + y^2$ is positive definite on $\R^2$ ; the forms $x^2 - y^2$ and $x^2$ are not. $Q$ is positive definite if and only if its associated symmetric bilinear form is an inner product.

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Vector space definition
Bilinear form definition
Quadratic form definition
Positive definite quadratic form definition

Tag	`00QN1N6MNT`
URI	`m2://00QN1N6MNT`
Kind	definition
Writing	W1 stated
MSC	15A63, 11E04
Defines	Positive definite quadratic form
Uses	Quadratic form

Positive definite quadratic form

Definition

Remarks

Reading path

Further reading

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