Differential and subdifferential properties of symplectic eigenvalues

Abstract

A real 2n × 2n matrix M is called a symplectic matrix if M T JM = J, where J is the 2n × 2n matrix given by J = ( O In −In O ) and In is the n × n identity matrix. A result on symplectic matrices, generally known as Williamson’s theorem, states that for any 2n × 2n positive definite matrix A there exists a symplectic matrix M such that M T AM = D ⊕ D where D is an n × n positive diagonal matrix with diagonal entries 0 < d1(A) ≤ · · · ≤ dn(A) called the symplectic eigenvalues of A. In this thesis, we study differentiability and analyticity properties of symplectic eigenvalues and corresponding symplectic eigenbasis. In particular, we prove that simple symplectic eigenvalues are infinitely differentiable and compute their first order derivative. We also prove that symplectic eigenvalues and corresponding symplectic eigenbasis for a real analytic curve of positive definite matrices can be chosen real analytically. We then derive an analogue of Lidskii’s theorem for symplectic eigenvalues as an application of our analysis. We study various subdifferential properties of symplectic eigenvalues such as Fenchel subdifferentials, Clarke subdifferentials and Michel-Penot subdifferentials. We show that symplectic eigenvalues are directionally differentiable and derive the expression of their first order directional derivatives.