Finite-dimensional points of continuity of Lat

Abstract

Associated with every linear transformation A on a finite-dimensional vector space V there is a collection Lat A of subspaces of V , namely the subspaces invariant under A . The collection lat A always contains the (trivial) subspace 0 and the (improper) subspace V . Since, moreover, it is closed under the formation of intersections and spans, it forms a lattice with respect to those operations (whence its name). The purpose of this paper is to interpret and to prove the following assertion: a necessary and sufficient condition that a linear transformation on a finite-dimensional complex vector space be a point of continuity of Lat is that it be non-derogatory . The assumption that the underlying coefficient field is the set of complex numbers is maintained throughout.