Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|cite|improve this question

1 Answer 1

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 $\begin{bmatrix}-A & B \\ B^{T},-D\end{bmatrix}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.