If $(\cdots A_{i_3}A_{i_2}A_{i_1})x_0\to 0$ for some $x_0$, what happens for any other $x$?

Question

Assume you have $m$ non-singular matrices $A_1,\dots,A_m$ and a given sequence of numbers $\{i_k\}_{k=1}^\infty$ for which $i_k\in\{1,\dots,m\}$. Also assume that you know that for some particular vector $x_0$ with $\|x_0\|=1$ you obtain $$ \lim_{N\to\infty}\left(A_{i_N}\cdots A_{i_1}\right)x_0=0 $$ My question is if you can conclude the same for any other vector $x$ with $\|x\|=1$ using the same sequence $\{i_k\}_{k=1}^\infty$.

Answer 1

There are simple counter-examples. Suppose $Ax_0=\frac 1 2 x_0$ and $Ay=y$ with $\|x\|=\|y\|=1$. Take $A_i=A$ for all $i$ and you have a counter-example.