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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

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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
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6. Williamson, J., Hadamards determinant theorem and the sum of four squares, Duke J. Math. 11 (1944), 65–81.Google Scholar