Proof of the Simon-Ando Theorem

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. \[ A = [ A 1 a m p ; E 1 E 2 a m p ; A 2 ] A = \begin {bmatrix} A_1&E_1\ E_2&A_2\end {bmatrix} \] where A 1 A_1 and A 2 A_2 are square blocks whose entries are all larger than those of E 1 E_1 and E 2 E_2 respectively. Setting \[ A k = [ A 1 ( k ) a m p ; E 1 ( k ) E 2 ( k ) a m p ; A 2 ( k ) ] , A^k=\begin {bmatrix} A^{(k)}_1&E^{(k)}_1\ E^{(k)}_2&A^{(k)}_2\end {bmatrix}, \] partitioned as in A 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 1 ( k ) A^{(k)}_1 to F 1 F_1 and A 2 ( k ) A^{(k)}_2 to F 2 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. \[ lim k → ∞ A k = [ α F 1 a m p ; β F 2 α F 1 a m p ; β F 2 ] \lim _{k\to \infty }A^k=\begin {bmatrix} \alpha F_1&\beta F_2\ \alpha F_1&\beta F_2\end {bmatrix} \] where α ≥ 0 \alpha \geq 0 and β ≥ 0 \beta \geq 0 . The purpose of this paper is to give a qualitative proof of the Simon-Ando theorem.