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 $\mu(G)$, $S$ is a real symmetric matrix supported on $G$ with nonnegative diagonal entries and whose negative index (i.e. number of negative eigenvalues) is denoted by $ \nu_{-}(S) $, then
$$ \nu_{-}(S)\leq \mu(G). $$
In particular, every $n\times n$ nonnegative tridiagonal stochastic matrix $P$ satisfies $ \nu_{-}(P)\leq \left\lfloor \frac{n}{2}\right\rfloor.$ Consequently, after ordering the eigenvalues of $P$ decreasingly, we have
$\lambda_{\lceil n/2\rceil}(P)\geq0,
\ \text{and hence} \
\lambda_2(P)\geq0, \mbox{ for } n\geq3. $ This gives an all-dimensional strengthening of the previously known $4\times4$ tridiagonal stochastic result. Finally, we show that this tridiagonal bound is sharp in every dimension and we explore some possible extension.
From: Bassam Mourad [view email]
[v1]
Fri, 19 Jun 2026 05:51:01 UTC (20 KB)
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