Characteristic polynomial/Direct sum decomposition/Fact

{{ Mathematical text/Fact |Text= {{ Factstructure|typ= |Situation= Let ${\displaystyle {}V}$ be a finite-dimensional ${\displaystyle {}K}$-vector space, and

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

be a linear mapping. |Condition= Let

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

denote a direct sum decomposition in ${\displaystyle {}\varphi }$-invariant linear subspaces. |Segue= |Conclusion= Then the characteristic polynomial fulfills the relation {{ Relationchain/display | \chi_{ \varphi } || {{op:Characteristic polynomial|\varphi {{|}}_U |}} \cdot {{op:Characteristic polynomial|\varphi{{|}}_W |}} || || || |pm=. }} |Extra= }} |Textform=Fact |Category= }}