Numerical enclosure for multiple eigenvalues of an Hermitian matrix whose graph is a tree |
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Authors: | Kenji Toyonaga |
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Institution: | aDepartment of Integrated Arts and Science, Kitakyushu National College of Technology, Fukuoka 802-0985, Japan |
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Abstract: | In this paper we consider a numerical enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. If an Hermitian matrix A whose graph is a tree has multiple eigenvalues, it has the property that matrices which are associated with some branches in the undirected graph of A have the same eigenvalues. By using this property and interlacing inequalities for Hermitian matrices, we show an enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. Since we do not generally know whether a given matrix has exactly a multiple eigenvalue from approximate computations, we use the property of interlacing inequalities to enclose some eigenvalues including multiplicities.In this process, we only use the enclosure of simple eigenvalues to enclose a multiple eigenvalue by using a computer and interval arithmetic. |
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Keywords: | Multiple eigenvalues Hermitian matrices Trees Validated bounds |
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