Endomorphism/Trigonalizable/Canonical additive decomposition/Fact/Proof

Proof

Due to fact, we have

$V=H_1\oplus \cdots \oplus H_m,$

where the $H_i$ are the generalized eigenspaces for the eigenvalues ${\lambda }_i$, and we have

$\phi ={\phi }_1\oplus \cdots \oplus {\phi }_m$

with ${\phi }_i=\phi {|}_{H_i}$. Let

$p_i:V⟶V$

denote the composition $V\to H_i\to V$, that is, $p_i$ is in particular a projection. We set

${\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}:={\lambda }_1{p}_1+\cdots +{\lambda }_m{p}_m.$

This mapping is obviously diagonalizable, on $H_i$ it is the multiplication with ${\lambda }_i$. Sei

${\phi }_{\mathrm{n}\mathrm{i}\mathrm{l}}:=\phi -{\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}.$

The property of this mapping of being nilpotent can be checked on the $H_i$ separately. There, we have

$(\phi -{\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}){|}_{H_i}={\phi }_i-\left({\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}\right){|}_{H_i}={\phi }_i-{\lambda }_i{\mathrm{Id}}_{H_i},$

so this is nilpotent. Moreover, ${\phi }_j$ and $p_i$ commute, since $p_i$ induces the identity on $H_i$ and on $H_j$, $j\ne i$, the zero mapping. Therefore, also the direct sums of those commute, and hence also $\phi$ and ${\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}$ commute. Thus, ${\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}$ and $\phi -{\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}={\phi }_{\mathrm{n}\mathrm{i}\mathrm{l}}$ commute.

To fact