On the simultaneous reduction of a pair of unitoid matrices to diagonal form

Let $A$ and $B$ be Hermitian $n\times n$ matrices with $A$ being nonsingular. According to a well-known theorem of matrix analysis, these matrices can be brought to diagonal form by one and the same Hermitian congruence transformation if and only if the matrix $C=A^{-1}B$ has a real spectrum and can be diagonalized by a similarity. An extension of this assertion to the case where two unitoid matrices are simultaneously reduced to diagonal form is stated and proved.