Let $F=\begin{bmatrix}A & B \\\\ B^{T} & D\end{bmatrix}$ be symmetric and strictly diagonally dominant (thus an $H$matrix). I also know that $B>0$ entrywise. What I am trying to show is that the Schur complements $F/A$ and $F/D$ are $M$matrices. It doesn't hold in the general case but my actual matrix is quite special and I'm sure it holds for it, after checking numerous examples. Still, the raw calculations with the standard formulas are too painful. So, does anybody know of a general condition I can check?
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.
If I understand correctly, this is not true. Suppose $F/A$ is an M matrix, then $F/A$ cannot be an M matrix. If $F$ satisfies your condition, so is $\begin{bmatrix}A & B \\ B^{T},D\end{bmatrix}$. 

