My question is identical to this question. It roughly asks:
If $A$ is symmetric (real and Hermitian) and positive definite ($x^TAx>0\;\forall x$) and $\hat{A}=A+\Delta A$ is a perturbed version of it, what condition(s) then ensure(s) that $\hat{A}$ is also positive-definite (but not necessarily symmetric)?
The answer to the mentioned question shows that if $$\|\Delta A\| < \frac{1}{\|A^{-1}\|},$$ where $\|\cdot\|$ is the induced norm from the Euclidean metric, then $\hat{A}$ is positive definite. However, this assumes that $\Delta A$ is symmetric.
I'm looking for references giving such conditions, but where $\Delta A$ can be arbitrary. I've looked at Weyl's inequality, but it again assumes that the perturbation is symmetric.