Endomorphism/K/Powers/Zero 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.

${}\varphi$ is asymptotically stable.
For every ${}v\in V$ , the sequence ${}\varphi ^{n}(v)$ , ${}n\in \mathbb {N}$ , converges to ${}0\in V$ .
There exists a generating system ${}v_{1},\ldots ,v_{m}\in V$ such that ${}\varphi ^{n}(v_{j})$ , ${}j=1,\ldots ,m$ , converges to ${}0$ .
The modulus of every complex eigenvalues of ${}\varphi$ is smaller than ${}1$ .