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Orthogonal matrices with zero diagonal
- Source :
- Canadian Journal of Mathematics / Journal Canadien de Mathématiques, 19, 1001-1010. Canadian Mathematical Society
- Publication Year :
- 1967
-
Abstract
- 1. Introduction. C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v = 2 (mod 4), and the skew C-matrices, of order v = 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, X)-systems on k and k + 1 in the sense of Bridges and Ryser [6]. Skew C-matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tourna­ ments introduced by Szekeres [24] and by Reid and Brown [21]. In § 4 we introduce the notion of negacyclic C-matrices, analogous to the similar notion introduced by Berlekamp in the setting of coding theory (cf. [4, p. 211]). Eigenvalues of negacyclic matrices are characterized and standard forms are obtained. Negacyclic C-matrices are interpreted in § 5 as the matrices of a special class of the relative difference sets introduced by Butson [7]. Exploiting some results of Elliott and Butson [10], we obtain a "multiplier theorem" for negacyclic C-matrices, and adapting a result of [2], we show that any negacyclic C-matrix has a nontrivial multiplier. Necessary conditions for the existence of a negacyclic C-matrix of order v are obtained in § 6. The nonexistence of nega­ cyclic C-matrices of all orders v ^ 226, v ^ 1 + Ph, with p prime, has been verified. This leads to the conjecture that they do not exist, unless v = 1 + PkPaley [19] constructed C-matrices of all orders v = 1 + pk, p prime. In § 7 it is shown that every Paley matrix is equivalent to a negacyclic C-matrix, a fact
- Subjects :
- Paley graph
Higher-dimensional gamma matrices
General Mathematics
010102 general mathematics
Conference matrix
01 natural sciences
Prime (order theory)
Combinatorics
Multiplier (Fourier analysis)
Paley construction
Matrix (mathematics)
0103 physical sciences
010307 mathematical physics
0101 mathematics
Eigenvalues and eigenvectors
Mathematics
Subjects
Details
- Language :
- English
- ISSN :
- 0008414X
- Database :
- OpenAIRE
- Journal :
- Canadian Journal of Mathematics / Journal Canadien de Mathématiques, 19, 1001-1010. Canadian Mathematical Society
- Accession number :
- edsair.doi.dedup.....4ca078978d274538fde3aeba531731cd