Let
$$P = \begin{bmatrix} A & B \\ B^\top & D\end{bmatrix}$$
where blocks $A$ and $D$ are positive definite. All the matrices $A$, $B$ and $D$ are $n \times n$.
I was wondering if one could actually derive a sufficient condition in terms of the minimum or maximum eigenvalues of the matrix $A$ such that $P >0$ can be ensured?
With the Schur complement, we see that $P > 0$ holds iff $D > 0$ and $A > BD^{-1}B^T$.
That said, we can guarantee that $P>0$ if $D > 0$ and $\lambda_{\min}(A) > \lambda_{\max}(BD^{-1}B^T)$. We could simplify this somewhat by noting that we have $$ \lambda_{\max}(BD^{-1}B^T) \geq \lambda_{\max}(B^TB)\lambda_{\max}(D^{-1}) = \frac{\lambda_{\max}(B^TB)}{\lambda_{\min}(D)}. $$