Normal forms of endomorphism-valued power series

Keane, Christopher; Szabó, Szilárd

doi:10.2140/involve.2018.11.81

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

Keywords

normal form, endomorphism, formal power series, Puiseux series

Mathematical Subject Classification 2010

Primary: 15A18, 15A21, 15A54

Secondary: 05E40

Milestones

Received: 17 July 2016

Revised: 31 August 2016

Accepted: 17 October 2016

Published: 17 July 2017

Communicated by Kenneth S. Berenhaut

Authors

Christopher Keane
Department of Mathematics Reed College 3203 SE Woodstock Blvd Portland, OR 97202 United States

Szilárd Szabó
Department of Mathematics Budapest University of Technology and Economics Egry J. u. 1, H ep. Budapest 1111 Hungary

Vol. 11, No. 1, 2018