Finite-dimensional vector space/Linear mapping/Two generalized eigenspaces/Fact/Proof

Proof

The characteristic polynomial of ${}\varphi$ has the form

${}\chi _{\varphi }=(X-\lambda )^{k}(X-\delta )^{\ell }F\,,$

where neither ${}\lambda$ nor ${}\delta$ is a zero of ${}F$. Because of fact, applied to ${}Q=(X-\delta )^{\ell }F$, we have

${}\operatorname {GeEig} _{\lambda }(\varphi )\cap \operatorname {kern} Q(\varphi )=0\,.$

Because of ${}\operatorname {GeEig} _{\delta }(\varphi )\subseteq \operatorname {kern} Q(\varphi )$, this implies immediately

${}\operatorname {GeEig} _{\lambda }(\varphi )\cap \operatorname {GeEig} _{\delta }(\varphi )=0\,.$

To fact