Conference matrix

Some authors use a more general definition, which requires there to be a single 0 in each row and column but not necessarily on the diagonal.

Thus, a normalized conference matrix has all 1's in its first row and column, except for a 0 in the top left corner, and is 0 on the diagonal.

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 squares;[7] there is a clever proof by elementary matrix theory in van Lint and Seidel.

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, 54, (not 58), 62 (sequence A000952 in the OEIS); for every one of these, it is known that a symmetric conference matrix of that order exists.

The essentially unique conference matrix of order 6 is given by All other conference matrices of order 6 are obtained from this one by flipping the signs of some row and/or column (and by taking permutations of rows and/or columns, according to the definition in use).

One conference matrix of order 10 is Skew-symmetric matrices can also be produced by the Paley construction.

This construction solves only a small part of the problem of deciding for which evenly even numbers n there exist skew-symmetric conference matrices of order n. Sometimes a conference matrix of order n is just defined as a weighing matrix of the form W(n, n−1), where W(n,w) is said to be of weight w > 0 and order n if it is a square matrix of size n with entries from {−1, 0, +1} satisfying W W T = w I.

For example, the matrix would satisfy this relaxed definition, but not the more strict one requiring the zero elements to be on the diagonal.

[11] As mentioned above, a necessary condition for a conference matrix to exist is that n−1 must be the sum of two squares.

Where there is more than one possible sum of two squares for n−1 there will exist multiple essentially different solutions for the corresponding conference network.