On the reduction of pairs of Hermitian or symmetric matrices to diagonal form by congruence (Q1076104)

Let A and B be complex \(n\times n\)-matrices, both Hermitian, or both symmetric, or one of each type. The authors study the transformation \((A,B)\to (TA\tilde T,TB\tilde T)\), where \(\tilde T=T^*\) if the corresponding A or B is Hermitian and \(\tilde T=T^ t\) if A or B is symmetric. The paper is devoted to the following problem: under what conditions on A and B a non-singular (unitary) matrix T can be found such that both \(TA\tilde T\) and \(TB\tilde T\) are diagonal ? All versions of the problem are solved if at least one of the matrices A and B is non- singular. If both are singular, the solutions are given for unitary T.