Lidskii theorem for directional derivatives of symplectic eigenvalues

Abstract

In this paper, we present a refinement of the symplectic Lidskii theorem for directional derivatives of symplectic eigenvalues, which is inspired by the work of Sendov [Electron. J. Linear Algebra 41(2025), 338-341]. We show that for a $2n \times 2n$ real positive definite matrix $A$ and symmetric matrices $H, K$ \begin{align*} d'(A;H+K)-d'(A;H) \prec_{π_A} d'(A;K). \end{align*} Here $\prec_{π_A}$ represents block-majorization corresponding to the partition $π_A$ of $\{1,\ldots, n\}$ given by the equality blocks of symplectic eigenvalues of $A$. We know that Hermitianization of a symmetric matrix occurs in the directional derivative expression of symplectic eigenvalues. We establish a componentwise inequality comparing the symplectic eigenvalues of a positive definite matrix with the ordinary eigenvalues of its associated Hermitianization, and also characterize the equality case. By using the aforementioned inequality, we then show that the symplectic Lidskii theorem follows from the directional derivative Lidskii theorem.