ThmDex – An index of mathematical definitions, results, and conjectures.

Finite real matrix is positive definite iff symmetric part is

Formulation 0

Let $A \in \mathbb{R}^{N \times N}$ be a D4571: Real matrix such that

(i)	$N \in 1, 2, 3, \ldots$ is a D5094: Positive integer

Then $A$ is a D4938: Positive definite real matrix if and only if $\frac{1}{2} (A + A^T)$ is.

Proofs

Proof 0

Let $A \in \mathbb{R}^{N \times N}$ be a D4571: Real matrix such that

(i)	$N \in 1, 2, 3, \ldots$ is a D5094: Positive integer

Result R3988: Expression for quadratic form of real square matrix in terms of symmetric part shows that \begin{equation} x^T A x = x^T \left( \frac{A + A^T}{2} \right) x \end{equation} for all $x \in \mathbb{R}^{N \times 1}$. The claim follows. $\square$