## 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) ×(n+1) nonnegative matrix whose nonzero eigenvalues are: λ_{0} ≥ |λ_{i}|, i = 1,..., r, r ≤ n, then for all x ≥ λo, (^{*}) Π^{r} _{i=0}(x-λ_{i}) ≤ x^{r+1}- λ^{r+1}_{0}. To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when 2(r + 1) ≥ (n+1), while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when n ≤ 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 (^{*}), from which it follows that the Boyle-Handelman conjecture is true. Actually, we do not start from the assumption that the λ_{i}'s are eigenvalues of a nonnegative matrix, but that λ_{1},..., λ_{r+1} satisfy λ_{0} ≥ |λ_{i}|, i = 1,..., r, and the trace conditions: (^{**}) Σ^{r} _{i=0}λ^{k}_{i} ≥ 0, for all k ≥ 1. 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,..., r. We further improve here on earlier results of the authors concerning this stronger form of the Boyle-Handelman conjecture.

Original language | English |
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Pages (from-to) | 1529-1538 |

Number of pages | 10 |

Journal | Proceedings of the American Mathematical Society |

Volume | 137 |

Issue number | 5 |

DOIs | |

State | Published - May 2009 |

## Keywords

- Characteristic polynomial
- Nonnegative matrices
- The inverse eigenvalue problem for nonnegative matrices