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| Titel: |
On a question of boyle and handelman concerning eigenvalues of nonnegative matrices |
| Auteur: |
Koltracht, I. Neumann, M. Xiao, D. |
| Verschenen in: |
Linear & multilinear algebra |
| Paginering: | Jaargang 36 (1993) nr. 2 pagina's 125-140 |
| Jaar: |
1993-11 |
| Inhoud: |
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 [image omitted] . 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 [image omitted] , 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 [image omitted] ? 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 [image omitted] , such that the inequality holds for all μ≥Cpμ1. Furthermore cp→ 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 [image omitted] , is an easY consequence of an additional observation by Keilson and Styan. |
| Uitgever: |
Taylor & Francis |
| Bronbestand: |
Elektronische Wetenschappelijke Tijdschriften |
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