Normal forms of endomorphism-valued power series

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 $gA{g}^{-1}$.