The uniform normal form of a linear mapping

TL;DR

This paper introduces a new normal form for linear mappings over finite-dimensional vector spaces that improves structural understanding and can be computed without root-finding, using only field operations.

Contribution

It presents a novel normal form for linear maps that surpasses the companion matrix in descriptive power and is computationally efficient over fields of characteristic zero.

Findings

Provides a normal form that enhances structural analysis of linear maps.

Allows computation using only field operations, avoiding polynomial root-finding.

Offers a more descriptive alternative to the companion matrix.

Abstract

Let $V$ be a finite dimensional vector space over a field $\mathrm{k}$ of characteristic $0$. Let $A$ be a linear mapping of $V$ into itself. This paper gives a normal form for $A$, which gives a better description of the structure of $A$ than the companion matrix. The computation of this normal form uses only operations from $\mathrm{k}$ and does not require finding roots of any polynomial.