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
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 correspondin...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2017-01-01
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| Series: | Special Matrices |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/spma-2017-0004 |
| _version_ | 1818717166385496064 |
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| author | Toyonaga Kenji Johnson Charles R. |
| author_facet | Toyonaga Kenji Johnson Charles R. |
| author_sort | Toyonaga Kenji |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-17T19:30:50Z |
| format | Article |
| id | doaj.art-e0432e8a72114a179774691e3418f49c |
| institution | Directory Open Access Journal |
| issn | 2300-7451 |
| language | English |
| last_indexed | 2024-12-17T19:30:50Z |
| publishDate | 2017-01-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Special Matrices |
| spelling | doaj.art-e0432e8a72114a179774691e3418f49c2022-12-21T21:35:16ZengDe GruyterSpecial Matrices2300-74512017-01-0151516010.1515/spma-2017-0004spma-2017-0004The classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the change in an edge valueToyonaga Kenji0Johnson Charles R.1Department of Integrated Arts and Science, Kitakyushu National College of Technology, Kokuraminami-ku, Kitakyushu, 802-0985, JapanDepartment of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, United States of AmericaWe 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.https://doi.org/10.1515/spma-2017-0004edgeseigenvaluesgraphmatrix entriesmultiplicityreal symmetric matrixtree15a1805c5015b5713h1505c05 |
| spellingShingle | Toyonaga Kenji Johnson Charles R. 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 Special Matrices edges eigenvalues graph matrix entries multiplicity real symmetric matrix tree 15a18 05c50 15b57 13h15 05c05 |
| title | 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 |
| title_full | 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 |
| title_fullStr | 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 |
| title_full_unstemmed | 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 |
| title_short | 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 |
| title_sort | classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the change in an edge value |
| topic | edges eigenvalues graph matrix entries multiplicity real symmetric matrix tree 15a18 05c50 15b57 13h15 05c05 |
| url | https://doi.org/10.1515/spma-2017-0004 |
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