Endomorphism/Trigonalizable/Direct sum/Fact

{{ Mathematical text/Fact |Text= {{ Factstructure|typ= |Situation= Let

${\displaystyle \varphi \colon V\longrightarrow V}$

be a trigonalizable ${\displaystyle {}K}$-endomorphism on the finite-dimensional ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$. |Condition=

|Segue= |Conclusion= Then ${\displaystyle {}V}$ is the direct sum of the generalized eigenspaces, that is,

${\displaystyle {}V=\operatorname {GeEig} _{\lambda _{1}}(\varphi )\oplus \cdots \oplus \operatorname {GeEig} _{\lambda _{m}}(\varphi )\,,}$

where ${\displaystyle {}\lambda _{1},\ldots ,\lambda _{m}}$ are the different eigenvalues of ${\displaystyle {}\varphi }$, and ${\displaystyle {}\varphi }$ is the direct sum of the restrictions {{ Mapping/display |name= \varphi_i = \varphi{{|}}_{H_i} | H_i | H_i || |pm= }} on the generalized eigenspaces. }} |Textform=Fact |Category= |Factname= |Request=Direct sum of the generalized eigenspaces }}