Endomorphism/Stable/Direct sum decomposition/Exercise

Let

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

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

${\displaystyle \varphi \colon V\longrightarrow V}$

be an endomorphism with the direct sum decomposition

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

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