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

• Autorzy: Kenji Toyonaga , Charles R. Johnson
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Edges; Eigenvalues; Graph; Matrix entries; Multiplicity; Real symmetric matrix; Tree

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2017
• Tom: 5
• Numer: 1
• Strony: 51-60

Daty

wydano

2017-01-01

otrzymano

2016-05-23

zaakceptowano

2016-09-27

online

2017-01-20

Twórcy

autor

Kenji Toyonaga

• Department of Integrated Arts and Science, Kitakyushu National College of Technology, Kokuraminami-ku, Kitakyushu, 802-0985,,

autor

Charles R. Johnson

• Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187-8795,,

Identyfikatory

DOI

10.1515/spma-2017-0004