Let $M$ to be a real-valued symmetric and positive-definite (PD) matrix (also sparse and banded if it helps)
$$ M= \begin{bmatrix} A & B\\ B^T & D \end{bmatrix} $$
Under what conditions the Schur complement of $M$ ( $S=D-B^T A^{-1} B$) is PD?
As far as I found, it holds if $M$ and $A$ are both PD. If this is true, how can say if $A$ is PD?
On
(In this answer, "$A \geq 0$" means $A$ is PSD and "$A > 0$" means $A$ is PD)
Let $f(u, v) = [u^T \ v^T]M[u^T \ v^T]^T$, where $M=\begin{bmatrix} A & B\\ B^T & D \end{bmatrix}.$ If $A>0$, one can show that $\inf_{u}f(u,v)=v^TSv$, where $S=D-B^T A^{-1} B.$
Theorem: if $A > 0$, then $M \geq 0$ iff $S \geq 0$.
Proof: If $S \geq 0$, then $$f(u,v) \geq \inf_{u}f(u,v)=v^TSv \geq 0\Rightarrow f(u,v) \geq 0$$ for all $u, v$, hence, $M \geq 0.$
If $M \geq 0$, then $$f(u,v) \geq 0 \Rightarrow \inf_{u}f(u,v) \geq 0 \Rightarrow v^TSv \geq 0$$ for all $v$, hence, $S \geq 0.$
Whatever it is that you're trying to do, it's helpful to understand the relationship between the Schur complement and the matrix $M$. Notably, we have (using block-matrix multiplication) $$ \pmatrix{I & 0\\-B^TA^{-1}&I}\pmatrix{A & B\\B^T & D}\pmatrix{I & 0\\-B^TA^{-1}&I}^T = \\ \pmatrix{A & 0\\0&D - B^TA^{-1}B} $$ Also, note that if $M$ is PD, then $A$ (which is a principal submatrix of $M$) must also be PD.