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

TL;DR

This paper investigates conditions under which a collection of Hermitian matrices can be simultaneously diagonalized via $^*$-congruence, providing theoretical equivalences and a practical numerical algorithm involving semidefinite programming and Jacobi-like methods.

Contribution

It offers new equivalent conditions for simultaneous diagonalization of Hermitian matrices via $^*$-congruence and introduces a combined SDP and Jacobi-like algorithm for practical computation.

Findings

Theoretical conditions for simultaneous diagonalization are established.

A semidefinite program can determine if matrices are simultaneously diagonalizable.

A numerical algorithm combining SDP and Jacobi-like steps effectively diagonalizes matrices.

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.