#### Vol. 11, No. 1, 2018

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Normal forms of endomorphism-valued power series

### Christopher Keane and Szilárd Szabó

Vol. 11 (2018), No. 1, 81–94
DOI: 10.2140/involve.2018.11.81
##### Abstract

We show for $n,k\ge 1$, and an $n$-dimensional complex vector space $V$ that if an element $A\in End\left(V\right)\left[\left[z\right]\right]$ has constant term similar to a Jordan block, then there exists a polynomial gauge transformation $g$ such that the first $k$ coefficients of $gA{g}^{-1}$ 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 $gA{g}^{-1}$.

##### Keywords
normal form, endomorphism, formal power series, Puiseux series
##### Mathematical Subject Classification 2010
Primary: 15A18, 15A21, 15A54
Secondary: 05E40