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

${}\varphi =\varphi _{1}\oplus \cdots \oplus \varphi _{m}\,$

with ${}\varphi _{i}=\varphi {|}_{H_{i}}$. Let

$p_{i}\colon V\longrightarrow V$

denote the composition ${}V\rightarrow H_{i}\rightarrow V$, that is, ${}p_{i}$ is in particular a projection. We set

${}\varphi _{\rm {diag}}:=\lambda _{1}p_{1}+\cdots +\lambda _{m}p_{m}\,.$

This mapping is obviously diagonalizable, on ${}H_{i}$ it is the multiplication with ${}\lambda _{i}$. Sei

${}\varphi _{\rm {nil}}:=\varphi -\varphi _{\rm {diag}}\,.$

The property of this mapping of being nilpotent can be checked on the ${}H_{i}$ separately. There, we have

${}{\left(\varphi -\varphi _{\rm {diag}}\right)}{|}_{H_{i}}=\varphi _{i}-{\left(\varphi _{\rm {diag}}\right)}{|}_{H_{i}}=\varphi _{i}-\lambda _{i}\operatorname {Id} _{H_{i}}\,,$

so this is nilpotent. Moreover, ${}\varphi _{j}$ and ${}p_{i}$ commute, since ${}p_{i}$ induces the identity on ${}H_{i}$ and on ${}H_{j}$, ${}j\neq i$, the zero mapping. Therefore, also the direct sums of those commute, and hence also ${}\varphi$ and ${}\varphi _{\rm {diag}}$ commute. Thus, ${}\varphi _{\rm {diag}}$ and ${}\varphi -\varphi _{\rm {diag}}=\varphi _{\rm {nil}}$ commute.

To fact