The classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the change in an edge value
The classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the change in an edge value
复制标题
边缘的分类以及边缘值变化导致的实对称矩阵特征值重数的变化
DOI：
10.1515/spma-2017-0004
复制
发表时间：
2017
期刊：
影响因子：
0.5
通讯作者：
Charles R. Johnson
中科院分区：
文献类型：
--
作者：
K. Toyonaga;Charles R. Johnson
摘要 我们给定一个实数对称矩阵 A，其图是一棵树 T，以及 A 的特征值及其重数。然后，当边缘被删除时（即 A 的相应条目被 0 替换），基于特定特征值的重数变化，T 的每条边缘可以被分类为四个类别之一。我们展示了边缘的每个可能分类的必要和充分条件。在 2-Parter 边、Parter 边和单 Parter 顶点之间观察到特殊关系。然后，我们根据边缘值的变化研究特征值重数的变化。我们展示了特征值的重数如何根据边缘的状态和边缘值而变化。这项工作解释了为什么在某些情况下边缘值对多重性没有影响。我们还更准确地描述了多重性如何随着两个相邻顶点的移除而变化。
Abstract We take as given a real symmetric matrix A, whose graph is a tree T, and the eigenvalues of A, with their multiplicities. Each edge of T may then be classified in one of four categories, based upon the change in multiplicity of a particular eigenvalue, when the edge is removed (i.e. the corresponding entry of A is replaced by 0).We show a necessary and suficient condition for each possible classification of an edge. A special relationship is observed among 2-Parter edges, Parter edges and singly Parter vertices. Then, we investigate the change in multiplicity of an eigenvalue based upon a change in an edge value. We show how the multiplicity of the eigenvalue changes depending upon the status of the edge and the edge value. This work explains why, in some cases, edge values have no effect on multiplicities. We also characterize, more precisely, how multiplicity changes with the removal of two adjacent vertices.