Conference matrix

From Wikipedia, the free encyclopedia

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 CTC is a multiple of the identity matrix I. Thus, if the matrix has order n, CTC = (n−1)I. 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[1][2].

Conference matrices arose in several independent ways. One original source is a problem in telephony[3]. Another is statistics [4]. Still another is elliptic geometry[5].

For n > 1, there are two kinds of conference matrix. Let us normalize C by, first (if the more general definition is used), rearranging the rows so that all the zeros are on the diagonal, and then negating any row or column whose first entry is negative. (These operations do 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, and is 0 on the diagonal. Let S be the matrix that remains when the first row and column of C are removed. Then either n is evenly even (a multiple of 4), and S is antisymmetric (as is the normalized C if its first row is negated), or n is oddly even (congruent to 2 modulo 4) and S is symmetric (as is the normalized C).

Contents

[edit] 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[3]; there is a clever proof by elementary matrix theory in van Lint and Seidel[5].

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, 54, 58, 62; for every one of these, it is known that a symmetric conference matrix of that order exists. Order 66 seems to be an open problem. (sequence A000952 in OEIS)[6]

[edit] Example

The essentially unique conference matrix of order 6 is given by

\begin{pmatrix}0 &+1 &+1 &+1 &+1& +1\\+1& 0 &+1 &-1 &-1& +1\\+1& +1& 0 &+1 &-1& -1\\+1& -1& +1& 0 &+1& -1\\+1& -1& -1& +1& 0& +1\\+1& +1& -1& -1& +1& 0 \end{pmatrix},

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).

[edit] 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 evenly even numbers n there exist antisymmetric conference matrices of order n.

[edit] Generalizations

Sometimes a conference matrix of order n is just defined as a weighting 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 Wt = w I.[2] Using this definition, the zero element is no more required to be on the diagonal, but it is easy to see that still there must be exactly one zero element in each row and column. For example, the matrix

\begin{pmatrix}1& 0& 1& 1\\0& -1& -1& 1\\ 1& -1& 0& -1\\ 1& 1& -1& 0 \end{pmatrix}

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

[edit] References

  1. ^ Malcolm Greig, Harri Haanpää, and Petteri Kaski, Journal of Combinatorial Theory, Series A, vol. 113, no. 4, 2006, pp 703-711, doi:10.1016/j.jcta.2005.05.005
  2. ^ a b Harald Gropp, More on orbital matrices, Electronic Notes in Discrete Mathematics, vol. 17, 2004, pp 179-183, doi:10.1016/j.endm.2004.03.036
  3. ^ a b Belevitch, V. (1950), Theorem of 2n-terminal networks with application to conference telephony. Electr. Commun., vol. 26, pp. 231-244.
  4. ^ Raghavarao, D. (1959). "Some optimum weighing designs". Annals of Mathematical Statistics 30 (2): 295–303. doi:10.1214/aoms/1177706253. MR0104322. 
  5. ^ a b van Lint, J.H., and Seidel, J.J. (1966), Equilateral point sets in elliptic geometry. Indagationes Mathematicae, vol. 28, pp. 335-348.
  6. ^ N.J.A. Sloane, OEIS, A000952.
  • Goethals, J.M., and Seidel, J.J. (1967), Orthogonal matrices with zero diagonal. Canadian Journal of Mathematics, vol. 19, pp. 1001-1010.
  • 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.