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Sets of matrices all infinite products of which converge. (English) Zbl 0746.15015
A set \(\sum=\{A_ i:i\geq 1\}\) of \(n\times n\) matrices is called an RCP set (right-convergent product set) if all infinite products with each element drawn from \(\sum\) converge. Such sets of matrices arise in constructing self-similar objects like von Koch’s snowflake curve, in various interpolation schemes, in constructing wavelets of compact support, and in studying nonhomogeneous Markov chains.
The paper gives necessary conditions and also some sufficient conditions for a set \(\sum\) to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in \(\sum\) and finite products of these matrices. Also finite RCP sets of column-stochastic matrices are completely characterized.

MSC:
15B57 Hermitian, skew-Hermitian, and related matrices
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
15B51 Stochastic matrices
41A05 Interpolation in approximation theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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