Before revision: This paper aims at giving some equivalent conditions for that a collection of finitely many of Hermitian matrices can be simultaneously diagonalizable via congruence (SDC) by a nonsingular matrix.
After revision: This paper aims at giving some equivalent conditions for that a collection of finitely many (not need pairwise commute) Hermitian matrices. Theoretically, we provide some equivalent conditions for that such a matrix collection can be simultaneously diagonalizable via congruence (SDC) by a nonsingular matrix.