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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2012.00598 |
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| _version_ | 1866911051693424640 |
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| author | Bui, Vuong |
| author_facet | Bui, Vuong |
| contents | 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 \[
\max_i \sqrt[m_i]{\max_{A_1,\dots,A_{m_i}\inΣ} (A_1\dots A_{m_i})_{i,i}} \le ρ(Σ) \le \max_i \sqrt[m_i]{\left(\frac{UD}{V}\right)^{3D^2} \max_{A_1,\dots,A_{m_i}\inΣ} (A_1\dots A_{m_i})_{i,i}}. \] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_00598 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A bound on the joint spectral radius using the diagonals Bui, Vuong Combinatorics 15A18, 65F15 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 \[ \max_i \sqrt[m_i]{\max_{A_1,\dots,A_{m_i}\inΣ} (A_1\dots A_{m_i})_{i,i}} \le ρ(Σ) \le \max_i \sqrt[m_i]{\left(\frac{UD}{V}\right)^{3D^2} \max_{A_1,\dots,A_{m_i}\inΣ} (A_1\dots A_{m_i})_{i,i}}. \] |
| title | A bound on the joint spectral radius using the diagonals |
| topic | Combinatorics 15A18, 65F15 |
| url | https://arxiv.org/abs/2012.00598 |