Sets of matrices all infinite products of which converge
An infinite product ∏∞i=1Mi of matrices converges (on the right) if limi→∞ M1 ... Mi exists. A set ∑={Ai:i≥1}of n x n matrices is called an RCP set (right- convergent product set) if all infinite products with each element drawn from ∑ 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. This paper gives necessary conditions and also some sufficient conditions for a set ∑ to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in ∑ and finite products of these matrices. Necessary and sufficient conditions are given for a finite set ∑ to be an RCP set having a limit function M∑(d)=π∞i=1Adi , where d=(d1,...,dn,...), which is a continuous function on the space of all sequences d with the sequence topology. Finite RCP sets of column-stochastic matrices are completely characterized. Some results are given on the problem of algorithmically deciding if a given set ∑ is an RCP set. © 1992.
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- Numerical & Computational Mathematics
- 49 Mathematical sciences
- 40 Engineering
- 09 Engineering
- 08 Information and Computing Sciences
- 01 Mathematical Sciences
Citation
Published In
DOI
ISSN
Publication Date
Volume
Issue
Start / End Page
Related Subject Headings
- Numerical & Computational Mathematics
- 49 Mathematical sciences
- 40 Engineering
- 09 Engineering
- 08 Information and Computing Sciences
- 01 Mathematical Sciences