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# C-Matrix

A -matrix is a symmetric () or antisymmetric () (-1,0,1)-matrix with diagonal elements 0 and others that satisfies

 (1)

where is the identity matrix, is known as a -matrix (Ball and Coxeter 1987). There are two symmetric -matrices of order 2,

 (2)

and two antisymmetric -matrices of order 2,

 (3)

Further examples include

 (4) (5)

There are no symmetric -matrices of order 4 or 22 (Ball and Coxeter 1987, p. 309). The following table gives the number of -matrices of orders , 2, ....

 type OEIS counts symmetric matrix A086260 1, 2, 0, 0, 0, 384, 0, 0, ... antisymmetric matrix A086261 1, 2, 0, 16, 0, 0, 0, 30720, ... total A086262 1, 4, 0, 16, 0, 384, 0, 30720, ...

A -matrix of an odd prime power order may be constructed using a general method due to Paley (Paley 1933, Ball and Coxeter 1987).

(-1,0,1)-Matrix, Conference Graph

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## References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 308-309, 1987.Belevitch, V. "Conference Matrices and Hadamard Matrices." Ann. de la Société scientifique de Bruxelles 82, 13-32, 1968.Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626-630, 1972.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. "Conference Matrices and Paley Graphs." In Distance Regular Graphs. New York: Springer-Verlag, p. 10, 1989.Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, p. 689, 1996.Paley, R. E. A. C. "On Orthogonal Matrices." J. Math. Phys. 12, 311-320, 1933.Raghavarao, D. Constructions and Combinatorial Problems in Design of Experiments. New York: Dover, 1988.Sloane, N. J. A. Sequences A086260, A086261, and A086262 in "The On-Line Encyclopedia of Integer Sequences."

C-Matrix

## Cite this as:

Weisstein, Eric W. "C-Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/C-Matrix.html