Characteristic polynomial/Coprime factorization/Direct sum/Fact

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

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

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

Then we have the

direct sum decomposition

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

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

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

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

Proof, Write another proof