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.

Explanations

L1 human

A positive definite quadratic form is positive everywhere except at the origin. Its graph is a bowl with the lowest point at the origin. Such forms describe squared lengths for different inner products.