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Normal forms of endomorphism-valued power series

We show for $n,k\geq1$, and an $n$-dimensional complex vector space $V$ that if an element $A\in\text{End}(V)[[z]]$ has constant term similar to a Jordan block, then there exists a polynomial gauge transformation $g$ such that the first $k$ coefficients of $gAg^{-1}$ have a controlled normal form. Furthermore, we show that this normal form is unique by demonstrating explicit relationships between the first $nk$ coefficients of the Puiseux series expansion of the eigenvalues of $A$ and the entries of the first $k$ coefficients of $gAg^{-1}$.