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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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.

\begin{displaymath}A=\bmatrix A_1&E_1\\ E_2&A_2\endbmatrix\end{displaymath}

where $A_1$ and $A_2$ are square blocks whose entries are all larger than those of $E_1$ and $E_2$ respectively.

Setting

\begin{displaymath}A^k=\bmatrix A^{(k)}_1&E^{(k)}_1\\ E^{(k)}_2&A^{(k)}_2\endbmatrix,\end{displaymath}

partitioned as in $A$, 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. $A^{(k)}_1$ to $F_1$ and $A^{(k)}_2$ to $F_2$. 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.

\begin{displaymath}\lim_{k\to\infty}A^k=\bmatrix \alpha F_1&\beta F_2\\ \alpha F_1&\beta F_2\endbmatrix\end{displaymath}

where $\alpha\geq 0$ and $\beta\geq 0$.

The purpose of this paper is to give a qualitative proof of the Simon-Ando theorem.


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Additional Information

D. J. Hartfiel
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
Email: hartfiel@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03033-X
PII: S 0002-9939(96)03033-X
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