On simultaneous diagonalization of one Hermitian and one symmetric form (Q788068)

The authors show that if A,B are complex \(n\times n\) matrices such that A is symmetric and B is hermitian and positive definite, then there exists a non-singular matrix U such that \(\bar U^ tBU=I\) and \(U^ tAU\) is a diagonal matrix with non-negative entries. As the authors suggest, it is surprising that this result is not known, but they have failed to find it in the literature. In addition, some related actions of the real orthogonal group and equations involving the unitary group are studied.