GEOMETRIC MATRICES AND AN INEQUALITY FOR (0,1)-MATRICES

Let A be an m × n (0, 1)-matrix with row vectors R1,..., Rm and column vectors C1,..., Cn. If there exist integers α, β such that RiRj = α whenever Ri ≠ Rj and CiCj = β whenever Ci ≠ Cj, then A will be called geometric. (RiRj, CiCj are the usual dot products of the vectors involved.) The geometric matrices are classified, and it is shown that (apart from certain trivialities) every geometric matrix is based on a symmetric balanced incomplete block design. Assume that each column of A has a zero entry and that Ci ≠ Cj for some i and j. Under these assumptions it is shown that m · min{RiRj:Ri≠Rj}≤n · max{CiCj:Ci≠Cj}, and that equality occurs if and only if A is geometric. The results generalize a theorem of de Bruijn and Erdös concerning combinatorial designs. © 1990.