Positive semidefinite matrix

Definition

Let $A$ be an $n \times n$ complex Hermitian matrix. The matrix $A$ is positive semidefinite when $$x^* A x \ge 0 \qquad \text{for all } x \in \C^n.$$ For a real symmetric $A$, the condition is $x^T A x \ge 0$ for all $x \in \R^n$.

Remarks

$A$ is positive semidefinite if and only if all its eigenvalues are nonnegative. Each positive definite matrix is positive semidefinite. The matrix $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is positive semidefinite but not positive definite. The notation $A \succeq 0$ is common.

Explanations

L1 human

A positive semidefinite matrix never gives a negative value of $x^T A x$. It may give zero for some nonzero vectors. Covariance matrices and Gram matrices are always of this kind.