The combinatorial structure of generalized eigenspaces – from nonnegative matrices to general matrices
The Perron–Frobenius spectral theory of nonnegative matrices motivated an intensive study of the relationship between graph theoretic properties and spectral properties of matrices. While for about seventy years research focused on nonnegative matrices, in the past fifteen years the study has been e...
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| Vydáno v: | Linear algebra and its applications Ročník 302-303; s. 173 - 191 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.12.1999
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| ISSN: | 0024-3795, 1873-1856 |
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| Abstract | The Perron–Frobenius spectral theory of nonnegative matrices motivated an intensive study of the relationship between graph theoretic properties and spectral properties of matrices. While for about seventy years research focused on nonnegative matrices, in the past fifteen years the study has been extended to general matrices over an arbitrary field. One of the major original problems in this context is determining the relations between the matrix analytic height characteristic of a matrix and the graph theoretic level characteristic. In this article the history of this problem is reviewed, from its introduction for nonnegative matrices, through its complete solution for nonnegative matrices, to the solution of the generalized version of the problem for general matrices. |
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| AbstractList | The Perron–Frobenius spectral theory of nonnegative matrices motivated an intensive study of the relationship between graph theoretic properties and spectral properties of matrices. While for about seventy years research focused on nonnegative matrices, in the past fifteen years the study has been extended to general matrices over an arbitrary field. One of the major original problems in this context is determining the relations between the matrix analytic height characteristic of a matrix and the graph theoretic level characteristic. In this article the history of this problem is reviewed, from its introduction for nonnegative matrices, through its complete solution for nonnegative matrices, to the solution of the generalized version of the problem for general matrices. |
| Author | Hershkowitz, Daniel |
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| Cites_doi | 10.1016/S0024-3795(96)00589-7 10.1017/S0013091500021507 10.1016/0024-3795(89)90034-7 10.1016/0024-3795(93)90369-Y 10.1016/0024-3795(86)90313-7 10.1080/03081089108818054 10.1016/0097-3165(76)90078-9 10.1016/0024-3795(85)90233-2 10.1016/0024-3795(88)90234-0 10.1016/0024-3795(92)90335-8 10.1016/0024-3795(89)90394-7 10.1137/0602046 10.1006/jctb.1993.1063 10.1080/03081088908817937 10.1007/BF01818561 10.1007/BF02787184 10.1016/0024-3795(75)90050-6 10.1016/0024-3795(94)90408-1 10.1016/0024-3795(88)90019-5 |
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J. doi: 10.1007/BF02787184 – volume: 12 start-page: 281 year: 1975 ident: 10.1016/S0024-3795(99)00064-6_BIB21 article-title: Algebraic egienspaces of non-negative matrices publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(75)90050-6 – ident: 10.1016/S0024-3795(99)00064-6_BIB18 – volume: 212/213 start-page: 309 year: 1994 ident: 10.1016/S0024-3795(99)00064-6_BIB11 article-title: Paths in directed graphs and spectral properties of matrices publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(94)90408-1 – volume: 106 start-page: 5 year: 1988 ident: 10.1016/S0024-3795(99)00064-6_BIB13 article-title: On the generalized nullspace of M-matrices and Z-matrices publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(88)90019-5 |
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