Negative index, matchings, and nonnegative eigenvalues of tridiagonal stochastic matrices

Abstract

We study negative eigenvalues of $n\times n$ stochastic matrices whose off-diagonal support is constrained by a sparse graph. The main tool is a matching-based inertia principle: if $G$ is bipartite with matching number $μ(G)$, $S$ is a real symmetric matrix supported on $G$ with nonnegative diagonal entries and whose negative index (i.e. number of negative eigenvalues counted with their multiplicities) is denoted by $ν_{-}(S) $, then \[ ν_{-}(S)\leq μ(G). \] In particular, every $n\times n$ nonnegative tridiagonal stochastic matrix $P$ satisfies $ ν_{-}(P)\leq \left\lfloor \frac{n}{2}\right\rfloor. $ Consequently, after ordering the eigenvalues of $P$ in the decreasing order, we have $ λ_{\lceil n/2\rceil}(P)\geq0, \ \text{and hence} \ λ_2(P)\geq0, \mbox{ for } n\geq3. $ This gives an all-dimensional strengthening of the previously known $4\times4$ tridiagonal stochastic result. Next, we show that this tridiagonal bound is sharp in every dimension in both reducible and irreducible cases. Finally, we explore some possible extension and raise some open questions.