Maybee, John S.; Pullman, Norman J. Tournament matrices and their generalizations. I. (English) Zbl 0714.05041 Linear Multilinear Algebra 28, No. 1-2, 57-70 (1990). 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. Cited in 1 ReviewCited in 19 Documents MSC: 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C20 Directed graphs (digraphs), tournaments Keywords:complex matrix; eigenvalue; rank; tournament matrix; spectral properties PDFBibTeX XMLCite \textit{J. S. Maybee} and \textit{N. J. Pullman}, Linear Multilinear Algebra 28, No. 1--2, 57--70 (1990; Zbl 0714.05041) Full Text: DOI References: [1] Bondy J. A., Graph Theory with Applications (1977) [2] DOI: 10.1090/S0002-9904-1968-12079-8 · Zbl 0167.03002 · doi:10.1090/S0002-9904-1968-12079-8 [3] Brauer A., Linear Alg. and its Appl. 5 pp 311– (1972) [4] de Caen D., Amer. Math. Mon. 5 (1972) [5] DOI: 10.1137/0402005 · Zbl 0674.05051 · doi:10.1137/0402005 [6] Horn R., Matrix Analysis (1985) · Zbl 0576.15001 [7] DOI: 10.1137/1011004 · Zbl 0186.33503 · doi:10.1137/1011004 [8] Moon J. W., Holt (1968) [9] Moon J. W., Proc. IFIP Congress pp 219– [10] DOI: 10.1137/1012081 · Zbl 0198.03804 · doi:10.1137/1012081 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.