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C-Matrix


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 I is the identity matrix, is known as a C-matrix (Ball and Coxeter 1987). 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 C-matrices of orders n=1, 2, ....

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


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."

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C-Matrix

Cite this as:

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

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