# Irreducible non-singular M-matrices and complex numbers

It is well known that a non-singular M-matrix that is irreducible has a strictly positive inverse (all entries $>0$).

An M-matrix is a matrix that has eigenvalues with positive real part, and the off-diagonal entries are non-positive ($\leq 0$). M-matrices can be expressed as $\alpha I-P$ for some non-negative matrix $P$ and real $\alpha > 0$.

A matrix $A$ is irreducible iff there does not exist a permutation matrix $P$ such that $P^TAP = \left[ \begin{array}{cc} B & C\\ 0 & D \end{array}\right]$. There are many definitions for irreducibility of a matrix.

Consider the M-matrix $M=sI-L$, where $s > 0$ and $L$ is a symmetric semi-definite non-negative and irreducible matrix.

What happens if I consider $s \in \mathbb{C}$, with $real (s) > 0$? Can I claim that the real part of $(sI-L)^{-1}$ is also positive? Is there an extension of M-matrices for complex numbers?

I am admittedly at a real loss with this, any help would be much appreciated!

• It might help if you could say what an M-matrix is, as well as an irreducible matrix. – MTS Mar 7 '11 at 15:09
• MTS...I edited the original post. Thanks for the comment. – dan Mar 7 '11 at 15:21