Given a matrix $A \in \mathbb{R}^{n \times n}$, $A + A^T$ is positive (semi)definite, what is $A$?

Question

Given a matrix $A \in \mathbb{R}^{n \times n}$, $A + A^T$ is positive (semi)definite, what is $A$?

or more generally, what are the properties on $A$ such that it holds true?

We see that if $A$ is a diagonal matrix with non-negative diagonals, then it holds true. But is there a wider class of matrices such that this property holds?

Some thoughts:

$A + A^T$ PSD $\implies x^T (A + A^T) x \geq 0 \implies x^T Ax \geq x^T A^T x$...seems like we are not getting any where.

Answer 1

start with this...

$$ \left( \begin{array}{rr} 1 & 1000 \\ -1000 & 2 \end{array} \right) $$