Sets of matrices all infinite products of which converge Export

@article{Daubechies_92, abstract = {An infinite product [product operator][infinity]i=1Mi of matrices converges (on the right) if limi-->[infinity] M1 ... Mi exists. A set [summation operator]={Ai:i[greater-or-equal, slanted]1}of n x n matrices is called an RCP set (right- convergent product set) if all infinite products with each element drawn from [summation operator] 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 [summation operator] to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in [summation operator] and finite products of these matrices. Necessary and sufficient conditions are given for a finite set [summation operator] to be an RCP set having a limit function M[summation operator](d)=[pi][infinity]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 [summation operator] is an RCP set.}, author = {Daubechies, Ingrid and Lagarias, Jeffrey C.}, citeulike-article-id = {3182610}, citeulike-linkout-0 = {http://dx.doi.org/10.1016/0024-3795(92)90012-Y}, citeulike-linkout-1 = {http://linkinghub.elsevier.com/retrieve/pii/0024-3795(92)90012-Y}, citeulike-linkout-2 = {http://www.sciencedirect.com/science/article/B6V0R-45D9TR7-2M/2/1335238ce894e4434dcd770ace33ce45}, comment = {Sets of matrices which all infinite products of them converge.}, day = {15}, doi = {10.1016/0024-3795(92)90012-Y}, journal = {Linear Algebra and its Applications}, keywords = {markov\_chains, matrix\_analysis}, month = {January}, pages = {227--263}, posted-at = {2008-09-02 20:19:05}, priority = {2}, title = {Sets of matrices all infinite products of which converge}, url = {http://dx.doi.org/10.1016/0024-3795(92)90012-Y}, volume = {161}, year = {1992} }

TY - JOUR ID - Daubechies_92 L3 - citeulike-article-id:3182610 N2 - An infinite product [product operator][infinity]i=1Mi of matrices converges (on the right) if limi-->[infinity] M1 ... Mi exists. A set [summation operator]={Ai:i[greater-or-equal, slanted]1}of n x n matrices is called an RCP set (right- convergent product set) if all infinite products with each element drawn from [summation operator] 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 [summation operator] to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in [summation operator] and finite products of these matrices. Necessary and sufficient conditions are given for a finite set [summation operator] to be an RCP set having a limit function M[summation operator](d)=[pi][infinity]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 [summation operator] is an RCP set. JF - Linear Algebra and its Applications EP - 263 TI - Sets of matrices all infinite products of which converge VL - 161 SP - 227 KW - markov_chains KW - matrix_analysis N1 - Sets of matrices which all infinite products of them converge. AU - Daubechies, Ingrid AU - Lagarias, Jeffrey PY - 1992/01/15/ UR - http://dx.doi.org/10.1016/0024-3795(92)90012-Y ER -