On the reduction of pairs of hermitian or symmetric matrices to diagonal form by congruence
Yoo Pyo Hong, Roger A. Horn, Charles R. Johnson · Linear Algebra and its Applications · 1986
Let A1, A2 be given n-by-n Hermitian or symmetric matrices, and consider the simultaneous transformations Ai→SAiS* if Ai is Hermitian or Ai→SAiST if Ai is symmetric. We give necessary and sufficient conditions for the existence of a unitary S which reduces both A1 and A2 to diagonal form in this way. When at least one of A1 or A2 is nonsingular, we give necessary and sufficient conditions for a reduction of this sort with a nonsingular S. These results are a generalization of the classical theorem from mechanics that a positive definite matrix and a Hermitian matrix can always be diagonalized simultaneously by a nonsingular congruence.