An Upper Bound on the Characteristic Polynomial of a Nonnegative Matrix Leading to a Proof of the Boyle–handelman Conjecture

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: λ0 ≥ |λi|, i = 1, . . . , r, r ≤ n, then for all x ≥ λ0, (∗) r ∏ i=0 (x− λi) ≤ x − λ 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 λi’s are eigenvalues of a nonnegative matrix, but that λ1, . . . , λr+1 satisfy λ0 ≥ |λi|, i = 1, . . . , r, and the trace conditions: (∗∗) r ∑ i=0 λi ≥ 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.