It is shown that for all nonzero n-component column vectors a and P over a field Fthere exists a set F of n-square matrices over F of cardinality n2 -2n+2 such that, for each n-square matrix A over F, A o=o or ATPB=p if and only if 1 is a characteristic value of PA for every P E r. Let F be a field with unit 1, and A an n-square matrix over F. If all row sums of A or all row sums of AT (the transpose of A) are 1, then A is a stochastic matrix. It is obvious that if A is stochastic then 1 is a characteristic value of PA for every n-square permutation matrix P. R. A. Brualdi and H. W. Wielandt [1] proved the converse. In this paper, it is shown as a corollary to a more general result that the n! permutation matrices can be replaced by a set of n-square matrices of cardinality n2-2n+2. Let Fn be the set of all n x 1 column vectors over F, and let a, f E Fn{O}. A set r of n-square matrices over F is called an (o, ,B)-set provided that, for each n-square matrix A over F, Aoa= a or ATP=p if and only if 1 is a characteristic'value of PA for every P E r. An (o, >)-set r is minimal if no proper subset of r is an (o, /l)-set. If cT'fl=0, then a and f are orthogonal. Let ek be the vector in Fn with all components zero, except component k, which is equal to 1. Denote the vector in Fn with all components equal to 1 by e. The characterization of stochastic matrices by Brualdi and Wielandt implies that the set of all n-square permutation matrices is an (e, e)-set of cardinality n!. We shall show that minimal (o, fl)-sets of cardinality n2-2n+2 can be easily constructed for all ., f E Fn-{O}. We first exhibit minimal (o, fl)-sets for two particular choices of a and fl. One of these is a = 9=e1, and the other is a =e1 and fl=8n LEMMA 1. For each positive integer n, the set of = {ns r m i u {o1ve + is a nI im j = 2, 3, * *-*ent of n-square matrices over F is a minimal (81, 81)-set. Presented to the Society November 17, 1973; received by the editors March 28, 1973. AMS (MOS) subject classifications (1970). Primary 15A18; Secondary 15A51. ? American Mathematical Society 1974