Proof of the Simon-Ando Theorem
Author: D. J. Hartfiel
Journal: Proc. Amer. Math. Soc. 124 (1996), 67-74
MSC (1991): Primary 15A51, 15A48
MathSciNet review: 1291772
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Abstract: In 1961, Simon and Ando wrote a classical paper describing the convergence properties of nearly completely decomposable matrices. Basically, their work concerned a partitioned stochastic matrix e.g.
where and are square blocks whose entries are all larger than those of and respectively.
partitioned as in , they observed that for some, rather short, initial sequence of iterates the main diagonal blocks tended to matrices all of whose rows are identical, e.g. to and to . After this initial sequence, subsequent iterations showed that all blocks lying in the same column as those matrices tended to a scalar multiple of them, e.g.
where and .
The purpose of this paper is to give a qualitative proof of the Simon-Ando theorem.
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D. J. Hartfiel
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
Keywords: Stochastic matrices, iterative behavior
Received by editor(s): February 9, 1994
Received by editor(s) in revised form: August 18, 1994
Communicated by: Joseph S. B. Mitchell
Article copyright: © Copyright 1996 American Mathematical Society