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 | References | Similar Articles | Additional Information

Abstract: In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if is an nonnegative matrix whose **nonzero** eigenvalues are: , , , then for all ,

To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when , while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when and when the spectrum of is real. They also showed that the conjecture is asymptotically true with the dimension.

Here we prove a slightly stronger inequality than in , from which it follows that the Boyle-Handelman conjecture is true. Actually, we do not start from the assumption that the 's are eigenvalues of a nonnegative matrix, but that satisfy , , and the trace conditions:

for all |

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 . 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

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.