On Convergence of Infinite Matrix Products with Alternating Factors from Two Sets of Matrices

We consider the problem of convergence to zero of matrix products AnBn⋯A1B1 with factors from two sets of matrices, Ai∈A and Bi∈B, due to a suitable choice of matrices {Bi}. It is assumed that for any sequence of matrices {Ai} there is a sequence of matrices {Bi} such that the corresponding matrix p...

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Bibliographic Details
Published in:Discrete dynamics in nature and society 2018-01, Vol.2018 (2018), p.1-5
Main Author: Kozyakin, Victor S.
Format: Article
Language:English
Subjects:
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Convergence
Inequality
Linear algebra
Norms
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Summary:We consider the problem of convergence to zero of matrix products AnBn⋯A1B1 with factors from two sets of matrices, Ai∈A and Bi∈B, due to a suitable choice of matrices {Bi}. It is assumed that for any sequence of matrices {Ai} there is a sequence of matrices {Bi} such that the corresponding matrix products AnBn⋯A1B1 converge to zero. We show that, in this case, the convergence of the matrix products under consideration is uniformly exponential; that is, AnBn⋯A1B1≤Cλn, where the constants C>0 and λ∈(0,1) do not depend on the sequence {Ai} and the corresponding sequence {Bi}. Other problems of this kind are discussed and open questions are formulated.
ISSN:1026-0226
1607-887X
DOI:10.1155/2018/9216760