The Class of Inverse M-Matrices Associated to Random Walks

Given $W=M^{-1}$, with $M$ a tridiagonal $M$-matrix, we show that there are two diagonal matrices $D,E$ and two nonsingular ultrametric matrices $U, V$ such that $DW\! E$ is the Hadamard product of $U$ and $V$. If $M$ is symmetric and row diagonally dominant, we can take $D=E=\mathbb{I}$. We relate this problem with potentials associated to random walks and study more closely the class of random walks that lose mass at one or two extremes.