利用对角线对联合谱半径的一个界
A bound on the joint spectral radius using the diagonals
AI总结：
本文基于非负矩阵有限集的对角元素，建立了联合谱半径的上下界，并通过与现有结果比较验证了该方法的有效性。
摘要：
本文的主要目的是基于非负矩阵有限集的对角元素，建立其联合谱半径的界。通过与该领域现有及相关结果进行比较，评估了该方法的有效性。具体而言，设 $Σ$ 为任意 $D\times D$ 非负矩阵的有限集，所有正元素的最大值为 $U$，最小值为 $V$。对于每个 $i=1,\dots,D$，设 $m_i$ 为满足存在 $A_1,\dots,A_{m_i}\inΣ$ 使得 $(A_1\dots A_{m_i})_{i,i} > 0$ 的任意数，若无此类矩阵则设 $m_i=1$。我们证明联合谱半径 $ρ(Σ)$ 的界为 \\[ \max_i \sqrt[m_i]{\max_{A_1,\dots,A_{m_i}\inΣ} (A_1\dots A_{m_i})_{i,i}} \le ρ(Σ) \le \max_i \sqrt[m_i]{\left(\frac{UD}{V}\right)^{3D^2} \max_{A_1,\dots,A_{m_i}\inΣ} (A_1\dots A_{m_i})_{i,i}}. \\]
英文摘要：
The primary aim of this paper is to establish bounds on the joint spectral radius for a finite set of nonnegative matrices based on their diagonal elements. The efficacy of this approach is evaluated in comparison to existing and related results in the field. In particular, let $Σ$ be any finite set of $D\times D$ nonnegative matrices with the largest value $U$ and the smallest value $V$ over all positive entries. For each $i=1,\dots,D$, let $m_i$ be any number so that there exist $A_1,\dots,A_{m_i}\inΣ$ satisfying $(A_1\dots A_{m_i})_{i,i} > 0$, or let $m_i=1$ if there are no such matrices. We prove that the joint spectral radius $ρ(Σ)$ is bounded by \[ \max_i \sqrt[m_i]{\max_{A_1,\dots,A_{m_i}\inΣ} (A_1\dots A_{m_i})_{i,i}} \le ρ(Σ) \le \max_i \sqrt[m_i]{\left(\frac{UD}{V}\right)^{3D^2} \max_{A_1,\dots,A_{m_i}\inΣ} (A_1\dots A_{m_i})_{i,i}}. \]
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
