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Does an irreducible $M$-matrix have positive diagonal entries?

An M-matrix $A$ is a matrix which can be expressed as $\alpha I−P$ for some non-negative matrix P and $\alpha\geq \rho(P)$ where $\rho(P)$ is the spectral radius of $P$.

A matrix $A$ is irreducible iff there does not exist a permutation matrix $P$ such that $P^{T}AP==\left( \begin{array}{cc} B & 0\\ C & D \\ \end{array} \right)$. There are many definitions for irreducibility of a matrix. (Reference: Chapter 1 in Nonnegative Matrices in the Mathematical Sciences)

Please help me to solve this or give me some suggestions to approach this problem.

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  • $\begingroup$ The structure of $A=\alpha I−P$ suggests that $A$ is irreducible if and only if $P$ is irreducible. Since $P$ is nonnegative and irreducible, I can use Perron-Frobenius Theorem. But I don't know how to use it for this problem. $\endgroup$ – La Rias Apr 13 '18 at 9:37
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As $\alpha I-P=(\alpha+1)\left[I-\frac1{\alpha+1}(I+P)\right]$, we may assume without loss of generality that $\alpha=1$, i.e. $A=I-P$ with $\rho(P)\le1$.

We claim that $A$ has a positive diagonal. Suppose the contrary that $a_{11}\le0$, i.e. $p_{11}\ge1$. Then the first diagonal entry of $P^k$ is also $\ge1$ for every positive integer $k$. Hence $\|P^k\|_\infty\ge1$ and Gelfand's formula implies that $\rho(P)\ge1$. Thus $\rho(P)=1$. As $P$ is irreducible, $v=Pv$ for some positive eigenvector $v$. Therefore $$ v_1=\sum_jp_{1j}v_j\ge v_1+\sum_{j>1}p_{1j}v_j, $$ meaning that all off-diagonal entries on the first row of $P$ are zero. This is a contradiction because $P$ is irreducible.

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  • $\begingroup$ Some small comments. I guess your equation should have p_1j instead of p_ij. Also the sum is over the index j and not p. $\endgroup$ – La Rias Apr 17 '18 at 13:18
  • $\begingroup$ @LaRias Yes, thanks for catching the typos. $\endgroup$ – user1551 Apr 17 '18 at 15:12
  • $\begingroup$ Is it possible to say anything about the converse, i.e., is it true that an M-matrix with positive diagonal entries is irreducible? $\endgroup$ – La Rias May 3 '18 at 7:52
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    $\begingroup$ @LaRias No. The identity matrix, for instance, is an M-matrix with a positive diagonal, but it is clearly reducible when its size is larger than 1. $\endgroup$ – user1551 May 3 '18 at 7:58

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