The centralizer of an endomorphism over an arbitrary field

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.