Proof of the Simon-Ando Theorem

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

Setting

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.

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

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