Characteristic polynomial/Coprime factorization/Direct sum/Fact

Let ${\displaystyle {}\varphi \colon V\rightarrow V}$ be a linear mapping on a finite-dimensional ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$, and let

${\displaystyle {}\chi _{\varphi }=P\cdot Q\,}$

be a factorization of the characteristic polynomial in coprime polynomials ${\displaystyle {}P,Q\in K[X]}$.

Then we have the

direct sum decomposition

${\displaystyle {}V=\operatorname {kern} P(\varphi )\oplus \operatorname {kern} Q(\varphi )\,,}$

where these linear subspaces are ${\displaystyle {}\varphi }$-invariant. We have

${\displaystyle {}\operatorname {kern} Q(\varphi )=\operatorname {Im} P(\varphi )\,,}$

and the restriction of ${\displaystyle {}P(\varphi )}$ onto ${\displaystyle {}\operatorname {kern} Q(\varphi )}$ is bijective.