Theoretical Statistics and Mathematics Unit, ISI Delhi

July 29, 2020 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Hemant Kumar Mishra,
ISI Delhi

Title:
Fenchel subdifferentials and first-order directional
derivatives of symplectic eigenvalues.

Abstract of Talk

For every $2n \times 2n$ positive definite matrix $A$ there exist $n$ positive numbers $d_1(A) \leq d_2(A) \leq \cdots \leq d_n(A)$ called the
symplectic eigenvalues of $A$. It is known that the symplectic
eigenvalue maps $d_1, d_2,...,d_n$ are continuous but not differentiable in general. We will show that the Fenchel subdifferential of the sum
of the symplectic eigenvalue maps $d_1+d_2+...+d_m$ exists for all $m \leq n$, and find their expressions. We also prove the existence of the first-order directional derivatives of the symplectic eigenvalue maps and derive their expressions.