# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## An upper bound on the characteristic polynomial of a nonnegative matrix leading to a proof of the Boyle-Handelman conjectureHTML articles powered by AMS MathViewer

by Assaf Goldberger and Michael Neumann
Proc. Amer. Math. Soc. 137 (2009), 1529-1538 Request permission

## Abstract:

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.

Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 15A48, 15A18, 11C08
• Retrieve articles in all journals with MSC (2000): 15A48, 15A18, 11C08
• 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
• 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