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?
Best Answer
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.