Endomorphism/Asymptotically stable/Direct sum decomposition/Exercise

Let

${}V=U\oplus W\,$

be a direct sum decomposition of a finite-dimensional ${}{\mathbb {K} }$-vector space ${}V$. Let

$\varphi \colon V\longrightarrow V$

be an endomorphism with the direct sum decomposition

${}\varphi =\psi \oplus \theta \,.$

Show that ${}\varphi$ is asymptotically stable if and only if ${}\psi$ ane ${}\theta$ are asymptotically stable.

Create a solution