Before revision:  As a consequence, this certainly solves such SDC-problem for collections of real symmetric matrices listed in J-B . Hiriart-Urruty, Potpourri of conjectures and open questions in nonlinear analysis and optimization,SIAM Review 49 (  2) , 2007 . Corresponding algorithms and illustrating examples by hand/coding \textsc{ are also presented.
After revision:  As a consequence, this certainly solves such SDC-problem for collections of real symmetric matrices listed in J-B . Hiriart-Urruty, Potpourri of conjectures and open questions in nonlinear analysis and optimization,SIAM Review 49 ( 1) a positive semidefinite program detecting whether the initial Hermitian matrices are simultaneously diagonalizable via *-congruence, and ( 2) , 2007 . Corresponding algorithms and illustrating examples by hand/coding \textsc{ are also presented.