The intersection of some classical equivalence classes of matrices

Abstract

Let A be an n X ra complex matrix. Let Sim (A) denote the similarity equivalence class of A, Conj(.4) denote the conjunctivity equivalence class of .4, UEquiv(.4) denote the unitaryequivalence equivalence class of .4, and 2/{{A) denote the unitary similarity equivalence class of A. Each of these equivalence classes has been studied for some time and is generally wellunderstood. In particular, canonical forms have been given for each equivalence class. Since the intersection of any two equivalence classes of .4 is again an equivalence class of .4, we consider two such intersections: CS(.4) = Sim(.4) fl Conj(.4) and UES(.4) = Sim(A) n UEquiv(.4). Though it is natural to first think that each of these is simply U{A), for each .4. we show by examples that this is not the case. We then try to classify which matrices .4 have CS(.4) = U{A). For matrices having CS(.4) ^ 1({A), we try to count the number of disjoint unitary similarity classes contained in CS(.4). Though the problem is not completely solved for CS(.4). we reduce the problem to non-singular, non-co-Hermitian matrices .4. A similar analysis is performed for UES(.4), and a (less simple) reduction of the problem is also achieved.