Differential and subdifferential properties of symplectic eigenvalues/ Hemant Kumar Mishra

Thesis (Ph.D.) - Indian Statistical Institute, 2021

Includes bibliographical references

Introduction -- 1 Preliminaries -- 2 Differentiability and analyticity of symplectic eigenvalues -- 3 First order directional derivatives of symplectic eigenvalues -- 4 Clarke and Michel-Penot subdifferentials of symplectic eigenvalues --

Guided by Prof. Tanvi Jain

A real 2n × 2n matrix M is called a symplectic matrix if MT 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 MT 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.