First order sensitivity analysis of symplectic eigenvalues

Abstract

For every $2n\times 2n$ positive definite matrix $A$ there are $n$ positive numbers $d_1(A)\le \dots \le d_n(A)$ 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.

arXiv comment24 pages

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Cited by in corpus (3)

• Block perturbation of symplectic matrices in Williamson's theorem
• Majorization in some symplectic weak supermajorizations
• Equality in some symplectic eigenvalue inequalities