Endomorphism/K/Powers/Convergence/Fact

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$ -vector space, and let

\varphi \colon V\longrightarrow V

be an endomorphism. Then the following properties are equivalent.

The sequence ${}\varphi ^{n}$ converges in ${}\operatorname {End} _{}{\left(V\right)}$ .
For every ${}v\in V$ , the sequence ${}\varphi ^{n}(v)$ , ${}n\in \mathbb {N}$ converges
There exists a generating system ${}v_{1},\ldots ,v_{m}\in V$ such that ${}\varphi ^{n}(v_{j})$ , ${}j=1,\ldots ,m$ , converges.
The modulus of every complex eigenvalue of ${}\varphi$ issmaller or equal ${}1$ , and if its modulus is ${}1$ , then the eigenvalue equals ${}1$ , and it is diagonalizable.
For a describing matrix ${}M$ of ${}\varphi$ , considered over ${}\mathbb {C}$ , the Jordan blocks of the Jordan normal form are
${\begin{pmatrix}\lambda &1&0&\cdots &0\\0&\lambda &1&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\\vdots &\cdots &0&\lambda &1\\0&\cdots &\cdots &0&\lambda \end{pmatrix}}$

with ${}\vert {\lambda }\vert <1$ , or equal ${}(1)$ .