Endomorphism/Generalized eigenspace/Algebraic multiplicity/Fact/Proof

Proof

We write the characteristic polynomial of ${}\varphi$ as

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

where ${}(X-\lambda )$ does not occur in ${}Q$ as a linear factor, that is, ${}k$ is the algebraic multiplicity of ${}\lambda$. Then, ${}P=(X-\lambda )^{k}$ and ${}Q$ are coprime, and, due to fact, we have the decompositon

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

and

$P(\varphi )={\left(\varphi -\lambda \operatorname {Id} \right)}^{k}\colon \operatorname {kern} Q(\varphi )\longrightarrow \operatorname {kern} Q(\varphi )$

is a bijection. Moreover,

${}H:=\operatorname {GeEig} _{\lambda }(\varphi )=\operatorname {kern} P(\varphi )\,,$

where the inclusion ${}\supseteq$ is clear, and the other inclusion follows from the fact that higher powers of ${}\varphi -\lambda \operatorname {Id}$ do not annihilate further elements, by the bijectivity on ${}\operatorname {kern} Q(\varphi )$ just mentioned. For the characteristic polynomial, we have, due to the direct sum decomposition according to fact, the relation

${}\chi _{\varphi }=\chi _{1}\cdot \chi _{2}\,,$

where ${}\chi _{1}$ is the characteristic polynomial of ${}\varphi {|}_{H}$ and ${}\chi _{2}$ is the characteristic polynomial of ${}\varphi {|}_{\operatorname {kern} Q(\varphi )}$. Since ${}(\varphi -\lambda \operatorname {Id} )^{k}$ restricted to ${}H$ is the zero mapping, the minimal polynomial of ${}\varphi {|}_{H}$ and, hence, also the characteristic polynomial ${}\chi _{1}$ are some power of ${}(X-\lambda )$, say

${}\chi _{1}=(X-\lambda )^{d}\,,$

where

${}d=\dim _{K}{\left(H\right)}\,.$

In particular, ${}d\leq k$, as ${}\chi _{1}$ is a divisor of ${}\chi _{\varphi }$. Assume that ${}d<k$. Then ${}\lambda$ is a zero of ${}\chi _{2}$ and ${}\lambda$ is an eigenvalue of ${}\varphi {|}_{\operatorname {kern} Q(\varphi )}$. But this is a contradiction to the fact that ${}P(\varphi )$ is a bijection on this space.

To fact