Vol. 16, No. 3, 1966

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Two inequalities in nonnegative symmetric matrices

David London

Vol. 16 (1966), No. 3, 515–536
Abstract

Marcus and Newman have made the following conjecture: Let A = (aij) be a n × n nonnegative symmetric matrix. Then

S (A )S(A2) ≦ nS (A3 ),

where

        n
∑
S(A) = i,j=1 aij.

After 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 = (aij) be a n × n nonnegative symmetric matrix. Then

   m    ∑n  m
S(A  ) ≦   si , m = 1,2,⋅⋅⋅ ,
i=1

where

    ∑n
si =   aij,  i = 1,⋅⋅⋅ ,n.
j=1

The 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.

Mathematical Subject Classification
Primary: 15.58
Milestones
Received: 8 October 1964
Published: 1 March 1966
Authors
David London