Two inequalities in nonnegative symmetric matrices
David London · Pacific Journal of Mathematics · 1966
Marcus and Newman have made the following conjecture: Let A = (α t y) be a n X n nonnegative symmetric matrix.Then S(A) S(A 2 ) ^ n S(A 3 ) ,whereAfter reducing the conjecture to a standard maximum problem of linear programming we prove that it holds for n £ 3. A counter example shows that for n ^ 4 the conjecture is wrong.We also consider the following conjecture: Let A = (α tJ ) be a n x n nonnegative symmetric matrix.ThenwhereThe validity of this conjecture is established in two cases:(1) m up to 5 and any n, (2) n up to 3 and any m.The general case remains open.We conclude this paper with two generalizations of the second theorem. NOTATION.Let A = (a i3 ) be a n x w real matrix.A is called nonnegative if a ti ^ 0, ΐ, j = 1, , n.The quadratic form corresponding to a symmetric A is denoted by A(x, x), that isHere (A#, a?) denotes, as usually, the scalar product of the real vectors x and Ax.Denote e = (1, , 1) and Ae = (s : , , s n ) = s = s(A).Si = Si(A) is thus the sum of the elements of the ith row of A. s = s(A) is ί/ie row sums vector of A. ^4 is generalized stochastic if A is nonnegative and if s(A) = cβ, where c is a scalar.Further