A note on finite determinacy of matrices

TL;DR

This paper establishes a precise criterion for when a 2x2 matrix over formal power series is finitely determined under a group action involving coordinate changes and invertible matrix multiplications.

Contribution

It provides a necessary and sufficient condition for finite G-determinacy of matrices in the specified algebraic setting.

Findings

Characterization of finite G-determinacy for 2x2 matrices.

Conditions involving group actions and formal power series.

Advances understanding of matrix classification under transformations.

Abstract

In this note, we give a necessary and sufficient condition for a matrix A in M to be finitely G-determined, where M is the ring of 2 x 2 matrices whose entries are formal power series over an infinite field, and G is a group acting on M by change of coordinates together with multiplication by invertible matrices from both sides.