Equality in some symplectic eigenvalue inequalities

TL;DR

This paper explores conditions for equality in symplectic eigenvalue inequalities, revealing parallels with classical eigenvalue inequalities for Hermitian matrices.

Contribution

It provides necessary and sufficient conditions for equality in symplectic analogs of Weyl's, Lidskii's, and Schur--Horn inequalities.

Findings

Equality conditions for symplectic Weyl's inequalities

Equality conditions for symplectic Lidskii's inequalities

Symplectic eigenvalue inequalities mirror classical eigenvalue inequalities

Abstract

In the last decade, numerous works have investigated several properties of symplectic eigenvalues. Remarkably, the results on symplectic eigenvalues have been found to be analogous to those of eigenvalues of Hermitian matrices with appropriate interpretations. In particular, symplectic analogs of famous eigenvalue inequalities are known today such as Weyl's inequalities, Lidskii's inequalities, and Schur--Horn majorization inequalities. In this paper, we provide necessary and sufficient conditions for equality in the symplectic analogs of the aforementioned inequalities. The equality conditions for the symplectic Weyl's and Lidskii's inequalities turn out to be analogous to the known equality conditions for eigenvalues.