Positive definite matrix

Definition

Let $A$ be an $n \times n$ complex Hermitian matrix. The matrix $A$ is positive definite when $$x^* A x > 0 \qquad \text{for all } x \in \C^n \text{ with } x \ne 0.$$ A real symmetric matrix $A$ is positive definite when $x^T A x > 0$ for all $x \in \R^n$ with $x \ne 0$.

Remarks

For a Hermitian $A$, the number $x^* A x$ is always real. The identity matrix and $\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$ are positive definite. $A$ is positive definite if and only if all its eigenvalues are positive. $A$ is positive definite if and only if $\langle x, y \rangle = y^* A x$ is an inner product on $\C^n$.

Explanations

L1 human

A positive definite matrix gives a positive number $x^T A x$ for every nonzero vector $x$. Such matrices behave like positive numbers in many ways: they have square roots and positive determinants. They appear as Hessians at minima and as covariance matrices.