Definite and indefinite quadratic forms

Summary

Let $A$ be a symmetric $n \times n$ matrix, and let $Q(x) = x^{\mathsf{T}} A x$ be its real quadratic form on $\mathbb{R}^n$. The form $Q$ is positive definite if $Q(x) > 0$ for all $x \ne 0$. It is negative definite if $Q(x) < 0$ for all $x \ne 0$. It is positive or negative semidefinite if $Q(x) \ge 0$ or $Q(x) \le 0$ for all $x$, and indefinite if it takes both positive and negative values.

Remarks

The form $Q$ is positive definite if and only if all eigenvalues of $A$ are positive. It is indefinite if and only if $A$ has a positive and a negative eigenvalue. On $\mathbb{R}^2$, $x_1^2 + x_2^2$ is positive definite, $x_1^2$ is positive semidefinite but not definite, and $x_1^2 - x_2^2$ is indefinite. In the second derivative test, the matrix $A$ is the Hessian matrix of a function at a critical point.

Explanations

L1 human

A positive definite form is a bowl: it is zero at the origin and positive everywhere else. An indefinite form is a saddle: it goes up in some directions and down in others.