Equivalent conditions for simultaneous diagonalization via ^*-congruence of Hermitian matrices

07/28/2020

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by   T. H. Le, et al.

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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. It surprisingly turns out that one of such equivalent conditions applies the semidefinite programming (SDP), which leads to a practical usefulness. 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 are also presented.