The centralizer of an endomorphism over an arbitrary field

TL;DR

This paper characterizes the centralizer of endomorphisms over arbitrary fields using generalized Jordan forms, providing formulas for its structure, dimension, and determinants, extending known results beyond classical cases.

Contribution

It introduces a comprehensive characterization of the centralizer for all minimal polynomials over any field, including separable and inseparable cases, using generalized canonical forms.

Findings

Derived formulas for the dimension of the centralizer.

Computed the structure of the centralizer over arbitrary fields.

Established methods to calculate determinants of elements in the centralizer.

Abstract

The centralizer of an endomorphism of a finite dimensional vector space is known when the endomorphism is nonderogatory or when its minimal polynomial splits over the field. It is also known for the real Jordan canonical form. In this paper we characterize the centralizer of endomorphisms over arbitrary fields for whatever minimal polynomial, and compute its dimension. The result is obtained via generalized Jordan canonical forms (for separable and non separable minimal polynomials). In addition, we also obtain the corresponding generalized Weyr canonical forms and the structure of its centralizers, which in turn allows us to compute the determinant of its elements.