Congruence of Hermitian matrices by Hermitian matrices (Q2370780)

The authors address the following interesting problem: Given two complex Hermitian \(n\)-by-\(n\) matrices \(A\) and \(B\), does there exist a non-singular Hermitian matrix of the same size, say \(C\), with \(B=CAC\)? Two particular cases are exhibited in detail: (i) \(A\) and \(B\) are two nonsingular simultaneously unitarily diagonalizable matrices; (ii) \(A\) and \(B\) are two nonsingular Hermitian \(2\)-by-\(2\) matrices. In both cases solutions to this problem of congruence by a Hermitian matrix are given. Also, it is shown that in either case real matrices \(A\) and \(B\) which are congruent by a Hermitian matrix are also congruent by a real symmetric matrix.