Jordan-block/2/Powers/Convergence behavior/Example

Let

${\displaystyle {}M={\begin{pmatrix}\lambda &1\\0&\lambda \end{pmatrix}}\,,}$

with ${\displaystyle {}\lambda \in \mathbb {C} }$. Then, according to exercise, we have

${\displaystyle {}M^{n}={\begin{pmatrix}\lambda ^{n}&n\lambda ^{n-1}\\0&\lambda ^{n}\end{pmatrix}}\,.}$

For

${\displaystyle {}\vert {\lambda }\vert <1\,,}$

this matrix sequence converges to the zero matrix, because every entry converges to ${\displaystyle {}0}$; for

${\displaystyle {}\lambda =1\,,}$

the sequence does not converge, because the entry in the right upper corner does converges; in fact, it is not even bounded. This is also true for ${\displaystyle {}\vert {\lambda }\vert \geq 1}$.