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On C-Matrices of Arbitrary Powers

Published online by Cambridge University Press:  20 November 2018

Richard J. Turyn
Richard J. Turyn*
Affiliation:
Raytheon Company, Sudbury, Massachusetts
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A C-matrix is a square matrix of order m + 1 which is 0 on the main diagonal, has ±1 entries elsewhere and satisfies . Thus, if , I + C is an Hadamard matrix of skew type [3; 6] and, if , iI + C is a (symmetric) complex Hadamard matrix [4]. For m > 1, we must have . Such matrices arise from the quadratic character χ in a finite field, when m is an odd prime power, as [χ(ai – aj)] suitably bordered, and also from some other constructions, in particular those of skew type Hadamard matrices. (For we must have m = a2 + b2, a, b integers.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 531 - 535
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Goethals, J. M. and Seidel, J. J., Orthogonal matrices with zero diagonal, Can. J. Math. 19 (1967), 1001–1010.Google Scholar
2. Goldberg, K., Hadamard matrices of order cube plus one, Proc. Amer. Math. Soc. 17 (1966), 744–746.Google Scholar
3. Hall, Marshall, Jr., Combinatorial theory (Blaisdell, Waltham, Mass., 1967).Google Scholar
4. Turyn, R., Complex Hadamard matrices, Combinatorial structures and their applications (Gordon and Breach, New York, 1970).Google Scholar
5. Wallis, J., On integer matrices obeying certain matrix equations, to appear in J. Combinatorial Theory.Google Scholar
6. Williamson, J., Hadamards determinant theorem and the sum of four squares, Duke J. Math. 11 (1944), 65–81.Google Scholar