There is a matrix as following,
\begin{eqnarray}
A = \left (
\begin{array}{l}
0 \quad \quad \quad \quad \quad \quad \quad ~~ 1\\
b \quad ~~~0 \quad \quad \quad \quad \quad a\\
ab \quad ~~ b \quad ~~~0\quad \quad ~a^2\\
\vdots \quad \quad~~~ \ddots ~~\ddots \quad \vdots\\
a^{n-2}b \quad \cdots ab \quad b \quad a^{n-1}
\end{array}
\right ),
\end{eqnarray}
where $A \in \mathbf{R^n}$, $a,b \in \mathbf{R},$and $0 <a < 1$, $|b|<1$. Then how to estimate the norm or the eigenvalue of $A$ and $A^k$, where $k \in \mathbf{N^{+}}$.
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