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| Titel: |
Tournament matrices and their generalizations, I. |
| Auteur: |
Maybee, John S. Pullman, Norman J. |
| Verschenen in: |
Linear & multilinear algebra |
| Paginering: | Jaargang 28 (1990) nr. 1-2 pagina's 57-70 |
| Jaar: |
1990-10 |
| Inhoud: |
If M is any complex matrix with rank (M + M* + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n - 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + MT + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n∉ {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented. |
| Uitgever: |
Taylor & Francis |
| Bronbestand: |
Elektronische Wetenschappelijke Tijdschriften |
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