Simultaneous diagonalization via congruence of Hermitian matrices: some equivalent conditions and a numerical solution (2007.14034v2)

Abstract: 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.