On a question of boyle and handelman concerning eigenvalues of nonnegative matrices

Abstract

Let A be an n × n nonnegative matrix whose eigenvalues are λ₁,…,λ_n, 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,…,λ_k 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,…,μ_p such that μ₁≥ max_1≤i≤p∣μ_i∣ 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 μ_i' are real, the fact that when , is an easY consequence of an additional observation by Keilson and Styan.

^∗Research supported in part by by NSF Grants DMS-9007030.

^†Research supported in part by by NSF Grants DMS-8901860 and DMS-9007030.

^‡Research supported in part by NSF Grants DMS-8901860 and DMS-9007030.

^∗Research supported in part by by NSF Grants DMS-9007030.

^†Research supported in part by by NSF Grants DMS-8901860 and DMS-9007030.

^‡Research supported in part by NSF Grants DMS-8901860 and DMS-9007030.

Notes

^∗Research supported in part by by NSF Grants DMS-9007030.

^†Research supported in part by by NSF Grants DMS-8901860 and DMS-9007030.

^‡Research supported in part by NSF Grants DMS-8901860 and DMS-9007030.