I think that it is true. That follows is a plan of attack for this purchase. According to the Schur theorem, we may assume that $A=[a_{i,j}]\in M_n(\mathbb{C})$ is upper-triangular. Let $f(n)$ be the integer (if it exists) s.t $A\in M_n(\mathbb{C})$, $\rho(A)\leq .9$, $||A||_2\leq 10$ and $k\geq f(n)$ imply that $||A^k||_2\leq 0.01$.

1. We consider matrices s.t. $|a_{i,i}|\leq 0.9$ or better $|a_{i,i}|= 0.9$ and s.t. $\rho(AA^*)=100$. We want that the $(A^k)$ are as big as possible; thus good candidates to approach the limit value $f(n)$ are matrices $A$ s.t. $a_{i,j}\geq 0$.

2. When $n=2,3$ the best candidates, in the previous context, are in the form $A=0.9 I_n+tJ_n$ where $J_n$ is the nilpotent Jordan block of dimension $n$. For $n=2$, we obtain $max(t)=9.919$ and (?) $f(2)=112$; for $n=3$, we obtain $max(t)=9.5$ and (?) $f(3)=181$.

3. I conjecture that, when $n>2$, the matrices that "realizes $f(n)$" have the above form. If it's true, then $f(n)\approx 70 n$. For instance, when $n=23$, $max(t)\approx 9.107$ and (?) $f(23)=1608$.

I think that it is true. That follows is a plan of attack for this purchase. According to the Schur theorem, we may assume that $A=[a_{i,j}]\in M_n(\mathbb{C})$ is upper-triangular. Let $f(n)$ be the integer (if it exists) s.t $A\in M_n(\mathbb{C})$, $\rho(A)\leq .9$, $||A||_2\leq 10$ and $k\geq f(n)$ imply that $||A^k||_2\leq 0.01$.

1. We consider matrices s.t. $|a_{i,i}|\leq 0.9$ or better $|a_{i,i}|= 0.9$ and s.t. $\rho(AA^*)=100$. We want that the $(A^k)$ are as big as possible; thus good candidates to approach the limit value $f(n)$ are matrices $A$ s.t. $a_{i,j}\geq 0$.

2. When $n=2,3$ the best candidates, in the previous context, are in the form $A=0.9 I_n+tJ_n$ where $J_n$ is the nilpotent Jordan block of dimension $n$. For $n=2$, we obtain $max(t)=9.919$ and (?) $f(2)=112$; for $n=3$, we obtain $max(t)=9.5$ and (?) $f(3)=181$.

3. I conjecture that, when $n>2$, the matrices that "realize $f(n)$" have the above form. If it's true, then $f(n)\approx 70 n$. For instance, when $n=23$, $max(t)\approx 9.107$ and (?) $f(23)=1608$.