On a strong form of a conjecture of Boyle and Handelman.

Goldberger, Assaf ; Neumann, Michael

The electronic journal of linear algebra, Tome 9 (2002), pp. 138-149.

Résumé

Summary: Let $\rho_{r, m}(x,\lambda):=(x - \lambda)^r\sum_{i=0}^m\binom{r+i-1}ix^{m-i}\lambda^i$. It is shown that if $\lambda_1,\dots, \lambda_n$ are complex numbers such that $\lambda_1=\lambda_2=\cdots=\lambda_r>0$ and $0\le\sum_{i=1}^n\lambda_i^k\le n\lambda_1^k$ for $1\le k\le m:=n-r$, then $$\prod_{i=1}^n(\lambda-\lambda_i)\le\rho_{r, m}(\lambda,\lambda_1), \quad{for all }\lambda\ge 6.75\lambda_1.\tag1$$ Moreover, if $r\ge m$, then (1) holds for all $\lambda\ge\lambda_1$, while if $r$, but $r$ is close to $m$, and $n$ is large, one can lower the constant of 6.75 in the inequality (1). The inequality (1) is inspired by, and related to, a conjecture of M. Boyle and D. E. Handelman [Ann. Math. (2) 133, No. 2, 249-316 (1991; Zbl 0735.15005)] on thenonzero spectrum of a nonnegative matrix.

Détail

Classification : 15A48, 15A18, 11C08
Keywords: nonnegative matrices, M-matrices, inverse eigenvalue problem