Special Families of Endomorphisms

Abstract

In this chapter we discuss special kinds of commuting families of endomorphisms of a finite dimensional vector space $V$. Important classes are families over which $V$ is a cyclic module, unigenerated families, commendable families, local families, dual families, and extended families. A particularly nice situation occurs when the commuting family ℱ contains a commendable endomorphism, since in this case the family is generated by that endomorphism and $V$ is a cyclic ℱ-module. More generally, commendable families are defined by the condition that their joint eigenspaces have the minimal possible dimension. One of the highlights of the chapter is the duality theorem which says that a family is commendable if and only if the dual vector space is a cyclic module over the dual family.