Lidskii theorem for directional derivatives of symplectic eigenvalues

Abstract

In this paper, we present a refinement of the symplectic Lidskii theorem for directional derivatives of symplectic eigenvalues, which is inspired by the work of Sendov [Electron. J. Linear Algebra 41(2025), 338-341]. We show that for a 2n × 2n real positive definite matrix A and symmetric matrices $H, K$ d'(A;H+K)−d'(A;H) ≺_{π_A} d'(A;K). Here ≺_{π_A} represents block-majorization corresponding to the partition π_A of {1,…, n} given by the equality blocks of symplectic eigenvalues of A. We know that Hermitianization of a symmetric matrix occurs in the directional derivative expression of symplectic eigenvalues. We establish a componentwise inequality comparing the symplectic eigenvalues of a positive definite matrix with the ordinary eigenvalues of its associated Hermitianization, and also characterize the equality case. By using the aforementioned inequality, we then show that the symplectic Lidskii theorem follows from the directional derivative Lidskii theorem.

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BibTeX RIS

BibTeX

@misc{mishra2026lidskii,
  title = {{Lidskii theorem for directional derivatives of symplectic eigenvalues}},
  author = {Hemant K. Mishra},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21620},
  eprint = {2609.21620},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21620},
}

RIS

TY  - UNPB
TI  - Lidskii theorem for directional derivatives of symplectic eigenvalues
AU  - Hemant K. Mishra
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21620
UR  - https://arxiv.org/abs/2609.21620
AB  - In this paper, we present a refinement of the symplectic Lidskii theorem for directional derivatives of symplectic eigenvalues, which is inspired by the work of Sendov [Electron. J. Linear Algebra 41(2025), 338-341]. We show that for a 2n × 2n real positive definite matrix A and symmetric matrices $H, K$ d'(A;H+K)−d'(A;H) ≺_π_A d'(A;K). Here ≺_π_A represents block-majorization corresponding to the partition π_A of {1,…, n} given by the equality blocks of symplectic eigenvalues of A. We know that Hermitianization of a symmetric matrix occurs in the directional derivative expression of symplectic eigenvalues. We establish a componentwise inequality comparing the symplectic eigenvalues of a positive definite matrix with the ordinary eigenvalues of its associated Hermitianization, and also characterize the equality case. By using the aforementioned inequality, we then show that the symplectic Lidskii theorem follows from the directional derivative Lidskii theorem.
ER  -

Abstract

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