---
tag: 0880F9EQCP
kind: definition
name: Positive semidefinite matrix
msc: [15B57, 15A63]
defines: 01MMVW8CKC
uses: [0EKZCX4K3P]
sources:
  - url: https://en.wikipedia.org/wiki/Definite_matrix
    locator: Definitions
---

## Definition

Let $A$ be an $n \times n$ complex [[Hermitian matrix|0P17NGQQMQ]].
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|0KX66CZMBB]] 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.