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@YemonChoi, Thanks for the comments. Please see my reply to Christian as well. I mainly just wanted to control $\|A^k\|$ in a non-asymptotic way. In other words, I'd like to have some version of non-asymptotic Gelfand's formula.
@ChristianRemling, Thanks for all your comments. I guess it's not clear what I want to prove and that's the beauty of its right? I just wanted to have something to bound about $\|A^k\|$ in terms of $\rho(A)^k$ weakly. For example, if $\rho(A) < 1/2$, can you say $\|A^k\| \le ||A||^{100} (2/3)^k$. I am looking for the correct theorem to be proved here -- so what's the right bound is just my question probably. Thanks!
@SebastianGoette Thanks! yeah I kind of thought such counter-examples. but this cases have a simple fix -- the frobenius norm of the matrix is bounded. So my point here is not to find a counter example but I'd like to get some constructive theory that really rule out the bad instances.
