Tridiagonal matrices with nonnegative entries

Abstract

In this paper we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let $d$ denote a nonnegative integer. Let $A$ denote a matrix in $\matR$ and let $\{\th_i\}_{i=0}^d$ denote the roots of the characteristic polynomial of $A$. We say $A$ is multiplicity-free whenever these roots are mutually distinct and contained in $\R$. In this case $E_i$ will denote the primitive idempotent of $A$ associated with $\th_i$ $(0 \leq i \leq d)$. We say $A$ is symmetrizable whenever there exists an invertible diagonal matrix $\Delta \in \matR$ such that $\Delta A \Delta^{-1}$ is symmetric. Let $\Gamma(A)$ denote the directed graph with vertex set $\{0,1,...,d\}$, where $i \rightarrow j$ whenever $i \neq j$ and $A_{ij} \neq 0$. Theorem: Assume that each entry of $A$ is nonnegative. Then the following are equivalent for $0 \leq s,t \leq d$: (i) The graph $\Gamma(A)$ is a bidirected path with endpoints $s$, $t$: (ii) The matrix $A$ is symmetrizable and multiplicity-free. Moreover the $(s,t)$-entry of $E_i$ times $(\th_i-\th_0)...(\th_i-\th_{i-1})(\th_i-\th_{i+1})...(\th_i-\th_d)$ is independent of $i$ for $0 \leq i \leq d$, and this common value is nonzero. Recently Kurihara and Nozaki obtained a theorem that characterizes the $Q$-polynomial property for symmetric association schemes. We view the above result as a linear algebraic generalization of their theorem.