I got this problem from Greenbaum's book of iterative methods. In page 14 he mentions that the 2-norm of matrix $A^k$ is asymptotically behaves like $v \left( \begin{array} { c } { k } \\ { j - 1 } \end{array} \right) [ \rho ( A ) ] ^ { k - j + 1 }$ when k is large enough.
i.e. $\left\| A ^ { k } \right\| \sim \nu \left( \begin{array} { c } { k } \\ { j - 1 } \end{array} \right) [ \rho ( A ) ] ^ { k - j + 1 }$ . Here $j$ is largest dimension of the diagonal submatrices $J_r$ of jordan form of $A$, $\rho \left( J _ { r } \right) = \rho ( A ) \text { and } \nu \text { is a positive constant. }$ I already figured out that for $\text { a } j - b y - j \text { Jordan block }J, \left\| J ^ { k } \right\| \sim \left( \begin{array} { c } { k } \\ { j - 1 } \end{array} \right) [ \rho ( J ) ] ^ { k - j + 1 } \text{ } \text{when}\quad k \rightarrow \infty$. How can I use that to show the bahavior of $\left\| A ^ { k } \right\|$? Any help would be really helpful.