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| Format: | Preprint |
| Published: |
2017
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1712.06356 |
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| _version_ | 1866915532767232000 |
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| author | Kozyakin, Victor |
| author_facet | Kozyakin, Victor |
| contents | We consider the problem of convergence to zero of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from two sets of matrices, $A_{i}\in\mathscr{A}$ and $B_{i}\in\mathscr{B}$, due to a suitable choice of matrices $\{B_{i}\}$. It is assumed that for any sequence of matrices $\{A_{i}\}$ there is a sequence of matrices $\{B_{i}\}$ such that the corresponding matrix product $A_{n}B_{n}\cdots A_{1}B_{1}$ converges to zero. We show that in this case the convergence of the matrix products under consideration is uniformly exponential, that is, $\|A_{n}B_{n}\cdots A_{1}B_{1}\|\le Cλ^{n}$, where the constants $C>0$ and $λ\in(0,1)$ do not depend on the sequence $\{A_{i}\}$ and the corresponding sequence $\{B_{i}\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1712_06356 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On convergence of infinite matrix products with alternating factors from two sets of matrices Kozyakin, Victor Optimization and Control Rings and Algebras 40A20, 39A22, 93B05, 93C55 We consider the problem of convergence to zero of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from two sets of matrices, $A_{i}\in\mathscr{A}$ and $B_{i}\in\mathscr{B}$, due to a suitable choice of matrices $\{B_{i}\}$. It is assumed that for any sequence of matrices $\{A_{i}\}$ there is a sequence of matrices $\{B_{i}\}$ such that the corresponding matrix product $A_{n}B_{n}\cdots A_{1}B_{1}$ converges to zero. We show that in this case the convergence of the matrix products under consideration is uniformly exponential, that is, $\|A_{n}B_{n}\cdots A_{1}B_{1}\|\le Cλ^{n}$, where the constants $C>0$ and $λ\in(0,1)$ do not depend on the sequence $\{A_{i}\}$ and the corresponding sequence $\{B_{i}\}$. |
| title | On convergence of infinite matrix products with alternating factors from two sets of matrices |
| topic | Optimization and Control Rings and Algebras 40A20, 39A22, 93B05, 93C55 |
| url | https://arxiv.org/abs/1712.06356 |