Classification of vertices and edges with respect to the geometric multiplicity of an eigenvalue in a matrix, with a given graph, over a field
Classification of vertices and edges with respect to the geometric multiplicity of an eigenvalue in a matrix, with a given graph, over a field
复制标题
在给定图的域上，根据矩阵中特征值的几何重数对顶点和边进行分类
DOI：
10.1080/03081087.2017.1389848
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发表时间：
2018
影响因子：
1.1
通讯作者：
K. Toyonaga
中科院分区：
文献类型：
--
作者：
Charles R. Johnson;Carlos M. Saiago;K. Toyonaga
摘要本文研究了在给定图G的一般域上矩阵a的非对角元素的已识别特征值的几何多重性。通过G的顶点（边）的分类，我们指的是当这个顶点（边）从G中移除以留下a的主子矩阵（修正）时的几何多重性的变化。对于厄米矩阵和树的情况，这种分类在确定给定图中矩阵之间特征值的可能多重性列表的问题中具有战略意义。在这里，我们对一般情况下的分类的看法和限定提供了一种工具，使过去的一些论点更加透明，并在经典和一般情况下提供了新的见解。给出了对角入口摄动下几何复数的稳定性和同伴顶点识别的一般下行机制的一些应用。
Abstract We are interested in the geometric multiplicity of an identified eigenvalue of a matrix A over a general field with a given graph G for its off-diagonal entries. By the classification of a vertex (edge) of G, we refer to the change in the geometric multiplicity of when this vertex (edge) is removed from G to leave a principal submatrix (modification) of A. Such classification in the case of Hermitian matrices and trees has been strategic in the problem of determining the possible lists of multiplicities for the eigenvalues among matrices with the given graph. Here, our view of, and qualification of, the classification in the general setting provides a tool that makes some past arguments more transparent and provides new insight in both the classical and general setting. Some applications are given to general downer mechanisms for the recognition of Parter vertices and to the stability of geometric multiplicity under perturbation of a diagonal entry.