Definition:Positive Semidefinite Matrix

Definition

Let $\mathbf A$ be a symmetric square matrix of order $n$.

Definition 1

$\mathbf A$ is positive semidefinite if and only if:

for all nonzero column matrices $\mathbf x$ of order $n$, $\mathbf x^\intercal \mathbf A \mathbf x$ is non-negative.

Definition 2

$\mathbf A$ is positive semidefinite if and only if:

all the eigenvalues of $\mathbf A$ are non-negative.

Also see

• Equivalence of Definitions of Positive Semidefinite Matrix

• Definition:Positive Definite Matrix

• Results about positive Semidefinite matrices can be found here.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): positive semidefinite