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, ....
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 Bruxelles82, 13-32, 1968.Brenner, J. and
Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math.
Monthly79, 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."
-matrix is a symmetric
(
) or antisymmetric
(
) 
(-1,0,1)-matrix with diagonal
elements 0 and others 
that satisfies
is the identity
matrix, is known as a 
-matrix
(Ball and Coxeter 1987). There are two symmetric 
-matrices of order 2,
![[0 -1; -1 0],[0 1; 1 0]](/images/equations/C-Matrix/NumberedEquation2.svg)
-matrices
of order 2,
![[0 1; -1 0],[0 1; -1 0].](/images/equations/C-Matrix/NumberedEquation3.svg)
![[0 + + +; - 0 - +; - + 0 -; - - + 0]](/images/equations/C-Matrix/Inline12.svg)
![[0 + + + + +; + 0 + - - +; + + 0 + - -; + - + 0 + -; + - - + 0 +; + + - - + 0].](/images/equations/C-Matrix/Inline15.svg)
-matrices
of order 4 or 22 (Ball and Coxeter 1987, p. 309). The following table gives
the number of 
-matrices
of orders 
,
2, ....
-matrix of an odd prime power order may
be constructed using a general method due to Paley (Paley 1933, Ball and Coxeter
1987).
