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,
| [画像: [0 -1; -1 0],[0 1; 1 0] ] |
(2)
|
![画像: [0 -1; -1 0],[0 1; 1 0]](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/C-Matrix/NumberedEquation2.svg){kind=link}
and two antisymmetric C-matrices of order 2,
| [画像: [0 1; -1 0],[0 1; -1 0]. ] |
(3)
|
![画像: [0 1; -1 0],[0 1; -1 0].](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/C-Matrix/NumberedEquation3.svg){kind=link}
Further examples include
![画像:[0 + + +; - 0 - +; - + 0 -; - - + 0]](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/C-Matrix/Inline12.svg){kind=link}
![画像:[0 + + + + +; + 0 + - - +; + + 0 + - -; + - + 0 + -; + - - + 0 +; + + - - + 0].](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/C-Matrix/Inline15.svg){kind=link}
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, ....
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 GraphExplore with Wolfram|Alpha
More things to try:
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."Referenced on Wolfram|Alpha
C-MatrixCite this as:
Weisstein, Eric W. "C-Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/C-Matrix.html