论文标题
使用对角线上的关节频谱半径上的一个
A bound on the joint spectral radius using the diagonals
论文作者
论文摘要
本文的主要目的是基于其对角线元素的有限的非负矩阵，在关节光谱半径上建立界限。与该领域的现有结果和相关结果相比，评估了这种方法的功效。特别是，让$σ$为$ d \ times d $的任何有限套件，非负矩阵，最大值$ u $，而最小的值$ v $比所有正分数。对于每个$ i = 1，\ dots，d $，令$ m_i $为任何数字，以便存在$ a_1，\ dots，a_ {m_i} \inς$满足$（a_1 \ dots a_ {m_i}）_ {m_i}）_ {m_i {m_i}）_ {i，i}> 0 $}> 0 $，或让$ m_i = 1 $ = 1 $我们证明关节频谱半径$ρ（σ）$由\ [ \ max_i \ sqrt [m_i] {\ max_ {a_1，\ dots，a_ {m_i} \inς}}（a_1 \ dots a_ {m_i}）_ {m_i}）_ {i} \ sqrt [m_i] {\ left（\ frac {ud} {v} \ right）^{3d^2} \ max_ {a_1，\ dots，a_ {m_i} \inς} \Inς}}（a_1 \ dots a_1 \ dots a___i} {m_i}） \]
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 \[ \max_i \sqrt[m_i]{\max_{A_1,\dots,A_{m_i}\inΣ} (A_1\dots A_{m_i})_{i,i}} \le ρ(Σ) \le \max_i \sqrt[m_i]{\left(\frac{UD}{V}\right)^{3D^2} \max_{A_1,\dots,A_{m_i}\inΣ} (A_1\dots A_{m_i})_{i,i}}. \]
