Let $A$ be real square matrix.

Let $\mathcal{F}(A)$ be the set of real matrices $A'$ of the same size such that $A'_{ii}=A_{ii}$ for all $i$, and for all $i,j$, $A_{ij}=0\Rightarrow A'_{ij}=0 \land A_{i,j} \ne 0 \implies A'_{i,j} \ne 0$: in other words, obtained from $A$ by modifying nonzero nondiagonal entries.

Q1 What are sufficient conditions for $A$ s.t. exists a positive definite $A'$ in $\mathcal{F}(A)$?

Since there are several definitions of positive definite, we don't require neither $A$ or $A'$ to be symmetric.

In practice this is possible for some $A$.

Partial answers are welcome. The definition of positive definite from mathworld

$M$ is positive definite if for all nonzero real vectors $x$, $x^T M x >0$.