CONFERENCE MATRIX

In mathematics, a 'conference matrix' (also called a ''C''-'matrix') is a square matrix ''C'' with 0 on the diagonal and +1 and −1 off the diagonal, such that ''C''^T''C'' is a multiple of the identity matrix ''I''. Thus, if the matrix has order ''n'', ''C''^T''C'' = (''n''−1)''I''.
Conference matrices arose in several independent ways. One original source is a problem in telephony (Belevitch 1950). Another is statistics (Raghavarao 1959). Still another is elliptic geometry (van Lint and Seidel 1966).
For ''n'' > 1, there are two kinds of conference matrices. Let us normalize ''C'' by negating any row or column whose first entry is negative. (This does not change whether a matrix is a conference matrix.) Thus, a normalized conference matrix has all 1's in its first row and column, except for a 0 in the top left corner. Let ''S'' be the matrix that remains when the first row and column of ''C'' are removed. Then either ''n'' is a multiple of 4, and ''S'' is antisymmetric (as is ''C'' if the first row is negated), or ''n'' is congruent to 2 (modulo 4) and ''S'' is symmetric (as is ''C'').

Contents

Symmetric conference matrices

Antisymmetric conference matrices

References

Symmetric conference matrices

If ''C'' is a symmetric conference matrix of order ''n'' > 1, then not only must ''n'' be congruent to 2 (mod 4) but also ''n'' − 1 must be a sum of two square integers (Belevich 1950; there is a clever proof by elementary matrix theory in van Lint and Seidel 1966).
Given a symmetric conference matrix, the matrix ''S'' can be viewed as the Seidel adjacency matrix of a graph. The graph has ''n'' − 1 vertices, corresponding to the rows and columns of ''S'', and two vertices are adjacent if the corresponding entry in ''S'' is negative. This graph is strongly regular of the type called (after the matrix) a conference graph.
The existence of conference matrices of orders ''n'' allowed by the above restrictions is known only for some values of ''n''. For instance, if ''n'' = ''q'' + 1 where ''q'' is a prime power congruent to 1 (mod 4), then the Paley graphs provide examples of symmetric conference matrices of order ''n'', by taking ''S'' to be the Seidel matrix of the Paley graph.
The first few possible orders of a symmetric conference matrix are ''n'' = 2, 6, 10, 14, 18, (not 22, since 21 is not a sum of two squares), 26, 30, (not 34 since 33 is not a sum of two squares), 38, 42, 46, 50; for every one of these, it is known that a symmetric conference matrix of that order exists. Order 54 seems to be an open problem. (See Sloane, Sequence A000952.)

Antisymmetric conference matrices

Antisymmetric conference matrices can also be produced by the Paley construction. Let ''q'' be a prime power with residue 3 (mod 4). Then there is a Paley digraph of order ''q'' which leads to an antisymmetric conference matrix of order ''n'' = ''q'' + 1. The matrix is obtained by taking for ''S'' the ''q'' × ''q'' matrix that has a +1 in position (''i,j'') and −1 in position (''j,i'') if there is an arc of the digraph from ''i'' to ''j'', and zero diagonal. Then ''C'' constructed as above from ''S'', but with the first row all negative, is an antisymmetric conference matrix.
This construction solves only a small part of the problem of deciding for which values of ''n'', multiples of 4, there exist antisymmetric conference matrices of order ''n''.

References

★ Belevitch, V. (1950), Theorem of 2''n''-terminal networks with application to conference telephony. Electr. Commun., vol. 26, pp. 231-244.

★ Goethals, J.M., and Seidel, J.J. (1967), Orthogonal matrices with zero diagonal. Canadian Journal of Mathematics, vol. 19, pp. 1001-1010.

★ van Lint, J.H., and Seidel, J.J. (1966), Equilateral point sets in elliptic geometry. Indagationes Mathematicae, vol. 28, pp. 335-348.

★ Raghavarao, D. (1959), Some optimum weighing designs. Annals of Mathematical Statistics, vol. 30, pp. 295-303.

★ Seidel, J.J. (1991), ed. D.G. Corneil and R. Mathon, Geometry and Combinatorics: Selected Works of J.J. Seidel. Boston: Academic Press. Several of the articles are related to conference matrices and their graphs.

★ Sloane, N.J.A., The On-Line Encyclopedia of Integer Sequences, Sequence A000952