Linear and multilinear algebra; matrix theory Definition

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.

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Matrix definition
Square matrix definition
Transpose definition
Conjugate transpose definition
Hermitian matrix definition
Positive semidefinite matrix definition

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Tag	`0880F9EQCP`
URI	`m2://0880F9EQCP`
Kind	definition
Writing	W1 stated
MSC	15B57, 15A63
Defines	Positive semidefinite matrix
Uses	Hermitian matrix

Positive semidefinite matrix

Definition

Remarks

Reading path

Where this leads

Further reading

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