关于Williamson定理到实对称矩阵的推广
On generalization of Williamson's theorem to real symmetric matrices
AI总结：
将Williamson定理从正定/半正定矩阵推广到一般实对称矩阵，引入实对称矩阵的辛特征值概念，给出显式构造和扰动界。
摘要：
Williamson定理指出：若$A$是$2n \times 2n$实对称正定矩阵，则存在$2n \times 2n$实辛矩阵$M$使得$M^T A M=D \oplus D$，其中$D$是对角元为正的$n \times n$对角矩阵，其对角元称为$A$的辛特征值。已知该定理可推广到核为$\mathbb{R}^{2n}$的辛子空间的$2n \times 2n$实对称半正定矩阵，此时$D$的部分对角元可为零。本文进一步将Williamson定理推广到$2n \times 2n$实对称矩阵，允许$D$的对角元为任意实数，从而将辛特征值的概念扩展到实对称矩阵。此外，我们给出了辛特征值的显式描述，构造了实现Williamson型分解的辛矩阵，并建立了对一类记作$\operatorname{EigSpSm}(2n)$的$2n \times 2n$实对称矩阵的辛特征值的扰动界。我们对$\operatorname{EigSpSm}(2n)$的辛特征值的扰动界推广了Bhatia和Jain \textit{[J. Math. Phys. 56, 112201 (2015)]}给出的正定矩阵辛特征值的已知扰动界。
英文摘要：
Williamson's theorem states that if $A$ is a $2n \times 2n$ real symmetric positive definite matrix then there exists a $2n \times 2n$ real symplectic matrix $M$ such that $M^T A M=D \oplus D$, where $D$ is an $n \times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of $A$. The theorem is known to be generalized to $2n \times 2n$ real symmetric positive semidefinite matrices whose kernels are symplectic subspaces of $\mathbb{R}^{2n}$, in which case, some of the diagonal entries of $D$ are allowed to be zero. In this paper, we further generalize Williamson's theorem to $2n \times 2n$ real symmetric matrices by allowing the diagonal elements of $D$ to be any real numbers, and thus extending the notion of symplectic eigenvalues to real symmetric matrices. Also, we provide an explicit description of symplectic eigenvalues, construct symplectic matrices achieving Williamson's theorem type decomposition, and establish perturbation bounds on symplectic eigenvalues for a class of $2n \times 2n$ real symmetric matrices denoted by $\operatorname{EigSpSm}(2n)$. Our perturbation bounds on symplectic eigenvalues for $\operatorname{EigSpSm}(2n)$ generalize known perturbation bounds on symplectic eigenvalues of positive definite matrices given by Bhatia and Jain \textit{[J. Math. Phys. 56, 112201 (2015)]}.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
