辛特征值方向导数的Lidskii定理
Lidskii theorem for directional derivatives of symplectic eigenvalues
- Indian Institute of Technology (ISM) Dhanbad（印度理工学院（ISM）丹巴德）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文受Sendov工作启发，细化了辛Lidskii定理，证明方向导数差满足块主化，并利用Hermitian化不等式推导出辛Lidskii定理。
AI中文摘要：
本文受Sendov [Electron. J. Linear Algebra 41(2025), 338-341]工作的启发，提出了辛Lidskii定理在辛特征值方向导数方面的精细化版本。我们证明，对于$2n \times 2n$实正定矩阵$A$以及对称矩阵$H, K$，有\begin{align*} d'(A;H+K)-d'(A;H) \prec_{\pi_A} d'(A;K). \end{align*}这里$\prec_{\pi_A}$表示对应于$\pi_A$划分的块主化，其中$\pi_A$是由$A$的辛特征值的相等块给出的$\{1,\ldots, n\}$的划分。我们知道，在辛特征值的方向导数表达式中会出现对称矩阵的Hermitian化。我们建立了一个分量不等式，将正定矩阵的辛特征值与其关联的Hermitian化的普通特征值进行比较，并刻画了等号成立的情形。通过利用上述不等式，我们进而证明辛Lidskii定理可由方向导数Lidskii定理推导得出。
英文摘要：
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 $2n \times 2n$ real positive definite matrix $A$ and symmetric matrices $H, K$ \begin{align*} d'(A;H+K)-d'(A;H) \prec_{π_A} d'(A;K). \end{align*} Here $\prec_{π_A}$ represents block-majorization corresponding to the partition $π_A$ of $\{1,\ldots, n\}$ 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.
