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!

  • $\begingroup$ It might help if you could say what an M-matrix is, as well as an irreducible matrix. $\endgroup$
    – MTS
    Mar 7, 2011 at 15:09
  • $\begingroup$ MTS...I edited the original post. Thanks for the comment. $\endgroup$
    – dan
    Mar 7, 2011 at 15:21

1 Answer 1


That will be H-matrix, which can be found at http://www.sciencedirect.com/science/article/pii/002437959300325T.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

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