An upper bound on the characteristic polynomial of a nonnegative matrix leading to a proof of the Boyle-Handelman conjecture
HTML articles powered by AMS MathViewer
- by Assaf Goldberger and Michael Neumann PDF
- 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.
References
- S. Ambikkumar and S. W. Drury, Some remarks on a conjecture of Boyle and Handelman, Linear Algebra Appl. 264 (1997), 63–99. MR 1465857, DOI 10.1016/S0024-3795(96)00402-8
- Jonathan Ashley, On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral, Linear Algebra Appl. 94 (1987), 103–108. MR 902070, DOI 10.1016/0024-3795(87)90081-4
- Mike Boyle and David Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. of Math. (2) 133 (1991), no. 2, 249–316. MR 1097240, DOI 10.2307/2944339
- M. Fiedler, Untitled private communication, 1982.
- Assaf Goldberger and Michael Neumann, On a strong form of a conjecture of Boyle and Handelman, Electron. J. Linear Algebra 9 (2002), 138–149. MR 1920915, DOI 10.13001/1081-3810.1082
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
- Julian Keilson and George P. H. Styan, Markov chains and $M$-matrices: inequalities and equalities, J. Math. Anal. Appl. 41 (1973), 439–459. MR 314873, DOI 10.1016/0022-247X(73)90219-9
- I. Koltracht, M. Neumann, and D. Xiao, On a question of Boyle and Handelman concerning eigenvalues of nonnegative matrices, Linear and Multilinear Algebra 36 (1993), no. 2, 125–140. MR 1308915, DOI 10.1080/03081089308818282
- I. G. Macdonald, Symmetric functions and orthogonal polynomials, University Lecture Series, vol. 12, American Mathematical Society, Providence, RI, 1998. Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, NJ. MR 1488699, DOI 10.1090/ulect/012
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
- 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
- Communicated by: Martin Lorenz
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2470809
Dedicated: Dedicated to the memory of our dear colleague Professor Israel Koltracht, 1949–2008