AN UPPER BOUND ON THE CHARACTERISTIC POLYNOMIAL OF A NONNEGATIVE MATRIX LEADING TO A PROOF OF THE BOYLE-HANDELMAN CONJECTURE

摘要

In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices,Boyle and Handelman conjectured that if A is all (n+1)x(n+1)nonnegative matrix whose nonzero eigenvalues are:lambda(0) >=vertical bar lambda(i)vertical bar,i=1,...,r,r<=n,then for allx>= lambda(0),(*)(r)Pi(i=0)(x-lambda(i))<=x(r+1)-lambda(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),whileKoltracht,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 lambda(i)'s are eigenvalues of a nonnegative matrix, but that lambda(1),...,lambda(r+1)satisfy lambda(0)>=vertical bar lambda(i)vertical bar,i=1,...,r,and the trace conditions:(**)(r)Sigma(i=0) lambda(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.