Simultaneous diagonalization via congruence of Hermitian matrices: some equivalent conditions and a numerical solution

2020/07/28 by Le, T. H., Nguyen, T. N.
#15A20 #15A22 #15B57 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2007.14034

Abstract

This paper aims at solving the Hermitian SDC problem, i.e., that of 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 Matlab are also presented.

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