  This paper aims at solving the Hermitian SDC problem, i.e., that of
\textit{simultaneously diagonalizing via $*$-congruence} 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 diagonalized via $^*$-congruence.% by a nonsingular matrix.
Interestingly, one of such conditions leads to the existence of a positive
definite solution to a semidefinite program (SDP). From practical point of
view, we propose an algorithm for numerically solving such problem. The
proposed algorithm is a combination of
  (1) a positive semidefinite program detecting whether the initial Hermitian
matrices are simultaneously diagonalizable via $*$-congruence, and
  (2) a Jacobi-like algorithm for simultaneously diagonalizing via
$*$-congruence the commuting normal matrices derived from the previous stage.
Illustrating examples by hand/coding in \textsc{Matlab} are also presented.
