Normal forms of endomorphism-valued power series

TL;DR

This paper establishes a method to normalize endomorphism-valued power series with a Jordan block structure using polynomial gauge transformations, ensuring a unique normal form linked to eigenvalue expansions.

Contribution

It introduces a procedure for normalizing such power series and proves the uniqueness of the normal form through explicit eigenvalue coefficient relationships.

Findings

Existence of polynomial gauge transformations to achieve controlled normal forms.

Uniqueness of the normal form based on eigenvalue Puiseux series.

Explicit formulas relating eigenvalue coefficients to matrix entries.

Abstract

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 $gAg^{-1}$.