Theoretical Statistics and Mathematics Unit, ISI Delhi

September 18, 2019 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Hemant K. Mishra,
ISI Delhi

Title:
Symplectic eigenvalues and a Lidskii type theorem

Abstract of Talk

With every 2n \times 2n positive definite matrix A there are n positive numbers associated, called the symplectic eigenvalues of A, and a basis of R^{2n} called a symplectic eigenbasis of A corresponding to these numbers. Symplectic eigenvalues have applications in different areas of Mathematics and Physics such as Symplectic Geometry, Quantum Information, Classical(Hamiltonian) Mechanics. In recent years, symplectic eigenvalues have gained much attention of physicists due to their important applications in Quantum Information. In this talk we shall discuss differentiability and analyticity of symplectic eigenvalues and symplectic eigenvectors, and compute their derivatives. Using our analysis we derive a symplectic analogue of the Lidskii's theorem (a result in classical Linear Algebra for eigenvalues of the sum of two Hermitian matrices) that gives a majorization relation among the symplectic eigenvalues of the sum of two matrices and that of the individual matrices.

This is a part of my Ph.D. research under the supervision of Prof. Tanvi Jain.