Variations on a theorem of Ryser

Abstract

A famous theorem of Ryser asserts that a v × v zero-one matrix A satisfying AAT = (k - λ)I + λJ with k ≠ λ must satisfy k + (v - 1)λ = k2 and ATA = (k - λ)I + λJ; such a matrix A is called the incidence matrix of a symmetric block design. We present a new, elementary proof of Ryser's theorem and give a characterization of the incidence matrices of symmetric block designs that involves eigenvalues of AAT. © Elsevier Science Inc., 1997.