On a question of boyle and handelman concerning eigenvalues of nonnegative matrices
I. Koltracht, Michael Neumann, D. Xiao · Linear and Multilinear Algebra · 1993
Let A be an n × n nonnegative matrix whose eigenvalues are λ1,…,λn, with λ1 its Perron root. Keilson and Styan and, more recently Ashley, have shown that . Boyle and Handelman in their paper on the inverse eigenvalue problem for nonnegative matrices ask whether this inequality can be strengthened as follows: Suppose λ1,…,λk are the nonzero eigenvalue of A, Then does the inequlity , hold? In this paper we partially answer their question by considering a set of p complex numbers (not necessarily the eigenvalues of a nonnegative matrix) μ1,…,μp such that μ1≥ max1≤i≤p∣μi∣ and such that ? Using Newton's identities for symmetric polynomials we show that the answer to this question is in the affirmative when p≤5. For p≥ 6, we show that there exists a constant , such that the inequality holds for all μ≥C p μ1. Furthermore c p → 1 as p→∞. Thus at least the same conclusions hold for the question posed by Boyle and Handelman. In the case when all the μi' are real, the fact that when , is an easY consequence of an additional observation by Keilson and Styan.