Positive semidefinite matrix

Definition · applies to symmetric matrices

Undergraduate

A symmetric $n \times n$ real matrix $A$ is positive semidefinite if $x^{\top} A x \geq 0$ for every vector $x \in \reals^n$.

Examples

The $3 \times 3$ identity matrix High school
The $3 \times 3$ identity matrix is positive definite, because $x^{\top} I x = \lVert x \rVert^2 > 0$ for every nonzero $x$. So it is also positive semidefinite.
The matrix $\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}$ Undergraduate
The matrix $\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}$ is positive definite, because its leading principal minors $4$ and $4 \cdot 3 - 2 \cdot 2 = 8$ are positive. So it is also positive semidefinite.

A symmetric matrix is positive semidefinite if and only if all of its eigenvalues are nonnegative.
Every matrix of the form $B^{\top} B$ is positive semidefinite, because $x^{\top} B^{\top} B x = \lVert B x \rVert^2 \geq 0$. Conversely, every positive semidefinite matrix can be written in this form.
A positive semidefinite matrix is positive definite if and only if it is invertible.
Sylvester’s criterion with $\geq$ does not test positive semidefiniteness: all leading principal minors of $\begin{bmatrix} 0 & 0 \\ 0 & -1 \end{bmatrix}$ are $\geq 0$, but $x = (0, 1)$ gives $x^{\top} A x = -1$. All principal minors (not only the leading ones) must be nonnegative.

Writing $A \succeq 0$ means that $A$ is positive semidefinite, and $A \succ 0$ means that $A$ is positive definite.

Positive definite vs. positive semidefinite: A symmetric matrix $A$ is positive semidefinite if $x^{\top} A x \geq 0$ for every vector $x$, and positive definite if $x^{\top} A x > 0$ for every nonzero $x$. Equivalently, the eigenvalues of $A$ are nonnegative or positive.