First order sensitivity analysis of symplectic eigenvalues

TL;DR

This paper investigates the sensitivity of symplectic eigenvalues of positive definite matrices, establishing the existence of directional derivatives and exploring subdifferential properties, thus advancing understanding of their mathematical behavior.

Contribution

It provides the first order sensitivity analysis of symplectic eigenvalues, including explicit formulas for directional derivatives and subdifferential characterizations.

Findings

Directional derivatives of symplectic eigenvalues exist.

Explicit formulas for derivatives are derived.

Subdifferential properties are analyzed.

Abstract

For every $2n \times 2n$ positive definite matrix $A$ there are $n$ positive numbers $d_1(A) \leq \ldots \leq d_n(A)$ associated with $A$ called the symplectic eigenvalues of $A.$ It is known that $d_m$ are continuous functions of $A$ but are not differentiable in general. In this paper, we show that the directional derivative of $d_m$ exists and derive its expression. We also discuss various subdifferential properties of $d_m$ such as Clarke and Michel-Penot subdifferentials.