Finite basic real matrix is positive definite iff symmetric part is

Let $A \in \mathbb{R}^{N \times N}$ be a D4571: Basic real matrix such that
 (i) $N \in 1, 2, 3, \ldots$ is a D5094: Positive basic integer
Then $A$ is a D4938: Positive definite basic real matrix if and only if $\frac{1}{2} (A + A^T)$ is.
Also known as
Characterisation of positive definiteness of basic real matrix by its symmetric part
Proofs

Let $A \in \mathbb{R}^{N \times N}$ be a D4571: Basic real matrix such that
 (i) $N \in 1, 2, 3, \ldots$ is a D5094: Positive basic integer
Result R3988: Expression for quadratic form of finite basic real square matrix in terms of symmetric part shows that $$x^T A x = x^T \left( \frac{A + A^T}{2} \right) x$$ for all $x \in \mathbb{R}^{N \times 1}$. The claim follows. $\square$