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

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

There is a matrix as following,

A=[0 0 0 1;b 0 0 a;ab b 0 $a^2$; $a^2b$ ab b $a^{N-1}$],

where $A \in \mathbf{R^4}$, $a,b \in \mathbf{R}$, and $|a|,|b|<1$ 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!

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

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

There is a matrix as following,

A=[0 0 0 1;b 0 0 a;ab b 0 $a^2$; $a^2b$ ab b $a^{N-1}$],

where $A \in \mathbf{R^4}$, $a,b \in \mathbf{R}$, and $|a|,|b|<1$ 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!