Endomorphism/Trigonalizable/Canonical additive decomposition/Fact/Proof

{{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= Due to fact, we have

${\displaystyle {}V=H_{1}\oplus \cdots \oplus H_{m}\,,}$

where the ${\displaystyle {}H_{i}}$ are the generalized eigenspaces for the eigenvalues ${\displaystyle {}\lambda _{i}}$, and we have

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

with {{ Relationchain | \varphi_i || \varphi{{|}}_{H_i} || || || |pm=. }} Let

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

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

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

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

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

The property of this mapping of being nilpotent can be checked on the ${\displaystyle {}H_{i}}$ separately. There, we have {{ Relationchain/display | { \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 } || || |pm=, }} so this is nilpotent. Moreover, ${\displaystyle {}\varphi _{j}}$ and ${\displaystyle {}p_{i}}$ commute, since ${\displaystyle {}p_{i}}$ induces the identity on ${\displaystyle {}H_{i}}$ and on ${\displaystyle {}H_{j}}$, ${\displaystyle {}j\neq i}$, the zero mapping. Therefore, also the direct sums of those commute, and hence also ${\displaystyle {}\varphi }$ and ${\displaystyle {}\varphi _{\rm {diag}}}$ commute. Thus, ${\displaystyle {}\varphi _{\rm {diag}}}$ and ${\displaystyle {}\varphi -\varphi _{\rm {diag}}=\varphi _{\rm {nil}}}$ commute. |Closure= }} |Textform=Proof |Category=See }}