There is a matrix as following, \begin{eqnarray} A=[0 0 0 1;b 0 0 a;ab b 0 $a^2$; $a^2$b ab b $a^{N-1}$], where $A \in \mathbf{R^4}$, $a,b \in \mathbf{R},$and $0 <a < 1$,$|b|<1$. Then how to estimate the norm or the eigenvalue of $A$ and $A^k$, where $k \in \mathbf{N^{+}}$. Furthermore, when $A \in \mathbf{R^n}$, then how to estimate.

Great thanks!