   ISSN 1088-6826(online) ISSN 0002-9939(print)

An upper bound on the characteristic polynomial of a nonnegative matrix leading to a proof of the Boyle-Handelman conjecture

Authors: Assaf Goldberger and Michael Neumann
Journal: Proc. Amer. Math. Soc. 137 (2009), 1529-1538
MSC (2000): Primary 15A48, 15A18, 11C08
DOI: https://doi.org/10.1090/S0002-9939-08-09701-3
Published electronically: January 2, 2009
MathSciNet review: 2470809
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In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if $A$ is an $(n+1)\times (n+1)$ nonnegative matrix whose nonzero eigenvalues are: $\lambda _0 \geq |\lambda _i|$, $i=1,\ldots ,r$, $r \leq \ n$, then for all $x \geq \lambda _0$, \begin{equation} \prod _{i=0}^{r} (x-\lambda _i) \leq x^{r+1}-\lambda _0^{r+1}.\tag *{$(\ast )$} \end{equation}

To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when $2(r+1)\geq (n+1)$, while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when $n\leq 4$ and when the spectrum of $A$ is real. They also showed that the conjecture is asymptotically true with the dimension.

Here we prove a slightly stronger inequality than in $(\ast )$, from which it follows that the Boyle–Handelman conjecture is true. Actually, we do not start from the assumption that the $\lambda _i$’s are eigenvalues of a nonnegative matrix, but that $\lambda _1,\ldots , \lambda _{r+1}$ satisfy $\lambda _0\geq |\lambda _i|$, $i=1,\ldots , r$, and the trace conditions: \begin{equation} \sum _{i=0}^{r} \lambda _i^k \geq 0, \ \mbox {for all} k \geq 1.\tag *{$(\ast \ast )$} \end{equation} A strong form of the Boyle–Handelman conjecture, conjectured in 2002 by the present authors, says that ($*$) continues to hold if the trace inequalities in ($**$) hold only for $k=1,\ldots ,r$. We further improve here on earlier results of the authors concerning this stronger form of the Boyle–Handelman conjecture.

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Assaf Goldberger
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Email: assafg@post.tau.ac.il

Michael Neumann
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269–3009
Email: neumann@math.uconn.edu

Keywords: Nonnegative matrices, the inverse eigenvalue problem for nonnegative matrices, characteristic polynomial
Received by editor(s): May 5, 2008
Published electronically: January 2, 2009
Additional Notes: The research of the second author was supported in part by NSA Grant No. 06G–232
Dedicated: Dedicated to the memory of our dear colleague Professor Israel Koltracht, 1949–2008
Communicated by: Martin Lorenz