矩阵与矩阵多项式的改进型包含-排除特征值定位集
Improved Inclusion-Exclusion Eigenvalue Localization Sets for Matrices and Matrix Polynomials
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中文总结 AI 辅助
本文基于S-严格对角占优框架提出新的包含-排除特征值定位集，统一并推广了Gershgorin与Dashnic-Zusmanovich结果，并扩展至矩阵多项式，数值实验证明其定位区域更锐利。
中文摘要 AI 辅助
近期研究表明，通过识别并排除那些保证不含特征值的子集，特征值定位区域往往可以得到显著锐化，由此引出了包含-排除定位集的概念。本文基于S-严格对角占优框架及相应的CKV定位区域，引入了一族新的包含-排除特征值定位集。对于指标集的任意非空子集S，我们构造了新的排除区域，并推导出相应的S-SDD包含-排除定位集。我们证明了复矩阵的每个特征值都属于所提出的区域，并通过对所有非空子集S对应的包含-排除区域取交集，定义了一个更精细的定位集。所提出的框架统一并推广了若干现有结果。特别地，我们表明经典的Gershgorin包含-排除定位集和Dashnic-Zusmanovich包含-排除定位集均可自然地作为我们构造的特例出现。因此，新的定位集通常能提供比这两种已知定位区域更锐利的特征值包含区域。该理论进一步推广至矩阵多项式，为多项式特征值问题提供了相应的包含-排除定位结果。多个数值算例验证了所提方法的有效性，并展示了相较于现有定位集所取得的改进。
英文摘要
Recent developments have shown that eigenvalue localization regions can often be significantly sharpened by identifying and excluding subsets that are guaranteed to contain no eigenvalues, leading to the concept of inclusion-exclusion localization sets. In this paper, we introduce a new family of inclusion-exclusion eigenvalue localization sets based on the S-strict diagonal dominance framework and the corresponding CKV localization regions. For an arbitrary nonempty subset S of the index set, we construct novel exclusion regions and derive the associated S-SDD inclusion-exclusion localization sets. We prove that every eigenvalue of a complex matrix belongs to the proposed regions and define a refined localization set obtained by intersecting the corresponding inclusion-exclusion regions over all nonempty subsets S. The proposed framework unifies and extends several existing results. In particular, we show that the classical Gershgorin inclusion-exclusion localization set and the Dashnic-Zusmanovich inclusion-exclusion localization set arise naturally as special cases of our construction. Consequently, the new localization set provides, in general, a sharper eigenvalue inclusion region than both of these previously known localization regions. The theory is further extended to matrix polynomials, yielding corresponding inclusion-exclusion localization results for polynomial eigenvalue problems. Several numerical examples illustrate the effectiveness of the proposed approach and demonstrate the improvement achieved over existing localization sets.
