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C-Matrix

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A C-matrix is a symmetric (C^T=C) or antisymmetric (C^T=-C) C_n (-1,0,1)-matrix with diagonal elements 0 and others +/-1 that satisfies

CC^T=(n-1)I,
(1)

where n is a positive integer and I is the identity matrix (Ball and Coxeter 1987). A C-matrix is also called a conference matrix. There are two symmetric C-matrices of order 2,

[0 -1; -1 0],[0 1; 1 0]
(2)

and two antisymmetric C-matrices of order 2,

[0 1; -1 0],[0 -1; 1 0].
(3)

Further examples include

C_4=[0 + + +; - 0 - +; - + 0 -; - - + 0]
(4)
C_6=[0 + + + + +; + 0 + - - +; + + 0 + - -; + - + 0 + -; + - - + 0 +; + + - - + 0].
(5)

There are no symmetric C-matrices of order 4 or 22 (Ball and Coxeter 1987, p. 309). The following table gives the number of distinct C-matrices of orders n=1, 2, .... Rows and columns are labeled, so matrices related by permutations or sign changes are counted separately. The order-one matrix (0) is both symmetric and antisymmetric, but is counted only once in the total.

typeOEIScounts
symmetric matrixA0862601, 2, 0, 0, 0, 384, 0, 0, ...
antisymmetric matrixA0862611, 2, 0, 16, 0, 0, 0, 30720, ...
totalA0862621, 4, 0, 16, 0, 384, 0, 30720, ...

A C-matrix of order q+1 may be constructed using a general method due to Paley whenever q is an odd prime power. The construction gives a symmetric C-matrix for q=1 (mod 4) and an antisymmetric C-matrix for q=3 (mod 4) (Paley 1933, Ball and Coxeter 1987).

Verma (2026) constructed a symmetric C-matrix of order 86, settling the previously open existence question for this order. The construction also yields ten pairwise nonisomorphic conference graphs with parameters (85,42,20,21).

See also

(-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."Verma, H. "A Symmetric Conference Matrix of Order 86." 6 Oct 2026. https://arxiv.org/abs/2610.08893.

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C-Matrix

Cite this as:

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

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