How to estimate the norm of a matrix

Question

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

Answer 1

You did not specify which norm, let me take the Frobenius norm, $||A||_F^2=\sum_{i,j=1}^n a_{ij}^2$, which gives $$||A||_F^2=\frac{\left(b^2-1\right) a^{2 n}+a^{2 n+2}-a^2 \left(b^2 n+1\right)+b^2 (n-1)+1}{\left(a^2-1\right)^2}.$$ For large $n$ and $|a|,|b|<1$ this tends to $||A||_F^2\rightarrow n\frac{b^2}{1-a^2}$.