Associated with every $2n\times 2n$ real positive definite matrix $A,$ there exist n positive numbers called the symplectic eigenvalues of $A,$ and a basis of $\mathbb {R}^{2n}$ called the symplectic eigenbasis of A corresponding to these numbers. In this paper, we discuss differentiability and analyticity of the symplectic eigenvalues and corresponding symplectic eigenbasis and compute their derivatives. We then derive an analogue of Lidskii’s theorem for symplectic eigenvalues as an application.

中文翻译：

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辛特征值的导数和 Lidskii 型定理

与每个 $2n\times 2n$ 实正定矩阵 $A,$ 相关联，存在n 个正数，称为 $A,$ 的辛特征值和 $\mathbb {R}^{2n}$ 的基，称为辛特征基的A对应于这些数字。在本文中，我们讨论了辛特征值和相应的辛特征基的可微性和解析性，并计算了它们的导数。然后，我们推导出辛本征值的 Lidskii 定理的类似物作为应用。