Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.

there exists universal constant $c > 0$ such that for any $n$ and any matrix $A$ with $\|A\|\le 10$ and $\rho(A) < 0.9$, it holds that for any $k > n^c$, $\|A^k\| \le .01$

Thanks!

(Note that this is closely related to a previously closed question. I guess now it is well-stated to be an non open-ended question)

Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.

There exists universal constant $c > 0$ such that for any $n$ and any matrix $A$ with $\|A\|\le 10$ and $\rho(A) < 0.9$, it holds that for any $k > n^c$, $\|A^k\| \le .01$

Thanks!

(Note that this is closely related to a previously closed question. I guess now it is well-stated to be an non open-ended question)