The intersection of some classical equivalence classes of matrices

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