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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
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
Published electronically: January 2, 2009
MathSciNet review: 2470809
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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 \vert\lambda_i\vert$, $ i=1,\ldots,r$, $ r \leq n$, then for all $ x \geq \lambda_0$,

$\displaystyle \prod_{i=0}^{r} (x-\lambda_i) \leq x^{r+1}-\lambda_0^{r+1}.$ $ (\ast)$

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 \vert\lambda_i\vert$, $ i=1,\ldots, r$, and the trace conditions:

$\displaystyle \sum_{i=0}^{r} \lambda_i^k \geq 0, \ $   for all$\displaystyle k \geq 1.$ $ (\ast\ast)$

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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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09701-3
PII: S 0002-9939(08)09701-3
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.