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Tournament matrices and their generalizations. I. (English) Zbl 0714.05041

Summary: 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 \(n\times n\) 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 marices M with 0 diagonal for which rank \((M+M^ T+I)=1\) and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible \(n\times n\) tournament matrices exist if and only if \(n\not\in \{2,3,4,5\}\) and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C20 Directed graphs (digraphs), tournaments
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References:

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