A note on perturbations of stochastic matrices (Q1841832)

Let \(A\) and \(B\) be \(n\) by \(n\) stochastic matrices, and \(A(t):=(1-t)A+tB\), where \(0\leq t\leq 1\). Thus \(A(t)\) is a stochastic matrix. The paper is on the existence and behavior of limit of \(A(t)^k\) when \(k\to\infty\). On the existence it shows that if \(P_A:=\lim_{k\to\infty}A^k\) or \(P_B:=\lim_{k\to\infty}B^k\) exists then so does \(P_{A(t)}:=\lim_{k\to\infty}A(t)^k\), and that when both \(P_A\) and \(P_B\) exist and \(AB=BA\), \(P_{A(t)}=P_AP_B\) for all \(t\) with \(0<t<1\). It also further studies the behavior of \(P_{A(t)}\) in more general settings when \(t\) or \(1-t\) is small.