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 $k \in \mathbf{N^{+}}$.

Great thanks!