Endomorphism/K/Power/Decomposition/Fact

Let ${\displaystyle {}V}$ be a finite-dimensional ${\displaystyle {}{\mathbb {K} }}$-vector space, and let

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

be a trigonalizable endomorphism, with the decomposition (in the sense of fact)

${\displaystyle {}\varphi =\varphi _{\rm {diag}}+\varphi _{\rm {nil}}\,,}$

with a diagonalizable and a nilpotent mapping, commuting with each other, and with

${\displaystyle {}\varphi _{\rm {nil}}^{k}=0\,.}$

Then the powers of ${\displaystyle {}\varphi }$ have the representation

${\displaystyle {}\varphi ^{n}=\varphi _{\rm {diag}}^{n}+n\varphi _{\rm {diag}}^{n-1}\circ \varphi _{\rm {nil}}+{\binom {n}{2}}\varphi _{\rm {diag}}^{n-2}\circ \varphi _{\rm {nil}}^{2}+{\binom {n}{3}}\varphi _{\rm {diag}}^{n-3}\circ \varphi _{\rm {nil}}^{3}+\cdots +{\binom {n}{k-1}}\varphi _{\rm {diag}}^{n-k+1}\circ \varphi _{\rm {nil}}^{k-1}\,.}$