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 \textsc{ are also presented.
This paper aims at solving the Hermitian SDC problem, i.e., that ofsimultaneously 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.
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giving some equivalent conditions for that
solving the Hermitian SDC problem, i.e., that of
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simultaneously diagonalizing via *-congruence
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of Hermitian matrices
(not need pairwise commute) Hermitian matrices. Theoretically, we provide some equivalent conditions for that such a matrix collection
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diagonalizable via congruence (SDC)
diagonalized via ^*-congruence.\%
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It surprisingly turns out that
Interestingly,
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equivalent conditions applies the semidefinite programming
conditions leads to the existence of a positive definite solution to a semidefinite program
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, which leads to a practical usefulness. As a consequence, this certainly solves such SDC-problem for collections of real symmetric matrices listed in
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J-B
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Hiriart-Urruty, Potpourri of conjectures and open questions in nonlinear analysis and optimization,
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SIAM Review
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49
From practical point of view, we propose an algorithm for numerically solving such problem. The proposed algorithm is a combination of
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1) a positive semidefinite program detecting whether the initial Hermitian matrices are simultaneously diagonalizable via *-congruence, and (
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, 2007
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. Corresponding algorithms and illustrating
a Jacobi-like algorithm for simultaneously diagonalizing via *-congruence the commuting normal matrices derived from the previous stage. Illustrating
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in
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Matlab