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.