The intersection of the similarity and conjunctivity equivalence classes Original Research Article

Let A be an n×n complex matrix. Let Sim(A) denote the similarity equivalence class ofA, let Conj(A) denote the conjunctivity equivalence class of A, and let image denote the unitary similarity equivalence class of A. Define CS(A)≡Sim(A)∩Conj(A). We seek to classify the matrices that have image, and show by an example that this is not true in general. But we show that it is true when A is Hermitian or is a scalar multiple of a Hermitian. For ngreater-or-equal, slanted3, we reduce the general n×n case to the case when A is non-singular and not a multiple of a Hermitian matrix. We completely classify the 2×2 case and find that image when A is non-singular or normal (or both).