Definition:Positive Definite Matrix

Definition

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

Definition 1

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

for all nonzero column matrices $\mathbf x$ of order $n$, $\mathbf x^\intercal \mathbf A \mathbf x$ is strictly positive.

Definition 2

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

all the eigenvalues of $\mathbf A$ are strictly positive.

Also known as

Some sources hyphenate positive definite: positive-definite.

Some sources refer to a positive definite matrix as symmetric positive definite, but under our definition the symmetric part of the definition is redundant.

Also see

• Equivalence of Definitions of Positive Definite Matrix

• Definition:Positive Semidefinite Matrix

• Results about positive definite matrices can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): positive definite
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): positive definite