Ritz Value Localization for Non-Hermitian Matrices
Ritz Value Localization for Non-Hermitian Matrices
复制标题
非厄米矩阵的 Ritz 值本地化
DOI：
10.1137/120872693
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Mark Embree
中科院分区：
文献类型：
--
作者：
Russell L. Carden;Mark Embree
埃尔米特矩阵的瑞利-里兹特征值估计服从柯西交错，这对理论、应用和算法都有有益的影响。相比之下，关于非厄米矩阵的 Ritz 值的结果很少，超出了它们在数值范围内的范围。为了表明这种 Ritz 值具有相当大的结构，我们在数值范围内建立了区域，其中必须包含一般矩阵的某些 Ritz 值。为了证明即使在极端的例子中也会出现局部化，我们仔细分析了三维 Jordan 块可能的 Ritz 值组合。
Rayleigh--Ritz eigenvalue estimates for Hermitian matrices obey Cauchy interlacing, which has helpful implications for theory, applications, and algorithms. In contrast, few results about the Ritz values of non-Hermitian matrices are known, beyond their containment within the numerical range. To show that such Ritz values enjoy considerable structure, we establish regions within the numerical range in which certain Ritz values of general matrices must be contained. To demonstrate that localization occurs even for extreme examples, we carefully analyze possible Ritz value combinations for a three-dimensional Jordan block.