## Oliver Club

Associated with every 2n×2n real positive definite matrix A, there exist n positive numbers called the symplectic eigenvalues of A. Symplectic eigenvalues have found applications in various areas of science, and more notably, in Gaussian quantum information theory. In the last decade, numerous works have investigated several properties of symplectic eigenvalues. Remarkably, the results on symplectic eigenvalues are 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 work, we provide necessary and sufficient conditions for equality in the symplectic analogs of the inequalities mentioned above. The equality conditions for the symplectic Weyl’s and Lidskii’s inequalities turn out to be analogous to the known equality conditions for eigenvalues.