The intersection of the similarity and conjunctivity equivalence classes (Q1863574)

Let \(M_n\) be the set of complex \((n\times n)\) matrices and \(\text{GL}_n\) be the set of \((n\times n)\) complex invertible matrices. Two matrices \(A,B\in M_n\) are similar if there exists a matrix \(S\in\text{GL}_n\) such that \(B= S^{-1}AS\). The matrices \(A,B\in M_n\) are conjunctive (or * congruent or Hermitian congruent) if there exists a matrix \(T\in\text{GL}_n\) such that \(B= T^* AT\), where \(T^*\) denotes the conjugate transpose of \(T\). Two matrices \(A,B\in M_n\) are unitarily similar if there exist the matrix \(U\in U_n\) (the set \(n\times n\) of unitary matrices \((U_n\subset\text{GL}_n)\)) such that \(B= U^*AU= U^{-1}AU\). We denote: the similarity equivalence class of \(A\in M_n\) by \(\text{Sim}(A)\); the cojunctive equivalence class of \(A\in M_n\) by \(\text{Conj}(A)\); and the unitary similarity equivalence class of \(A\in M_n\) by \(U(A)\). It is obvious that \(U(A)\subset \text{Sim}(A)\) and \(U(A)\subset \text{Conj}(A)\). Let \[ CS(A)= (\text{Sim}(A)\cap \text{Conj}(A)). \] In this article the following question is investigated: When the equality \(CS(A)= U(A)\) is actual for fixed \(A\in M_n\)? Let \(H_n\) be a set of Hermitian \(n\times n\) matrices. The next result is true: Theorem 1. If \(aA\in H_n\) for some non-zero complex number \(a\), then \(CS(A)= U(A)\). The main result of the paper is the following: Theorem 2. Let \(A_1\in \text{GL}_n\), and let \(m\) be a given nonnegative integer. Then \(B\in CS(A_1\oplus 0_m)\) if and only if \(B\) is unitarily similar to a matrix of the form \(B_1\oplus 0_m\), for some \(B_1\in CS(A_1)\). Furthermore, \(CS(A_1)= U(A_1)\) if and only if \(CS(A_1\oplus 0_m)= U(A_1\oplus 0_m)\).