ELA, Volume 3, pp. 119-128, July 1998, abstract.


A refinement of an inequality of Johnson, Loewy and London 
on nonnegative matrices and some applications.

Thomas J. Laffey and Eleanor Meehan


Let A be an entrywise nonnegative n x n matrix and let 
s_k := trace(A^k) (k=1,2,...).  It is shown that if n is odd and 
s_1=0, then (n-1)s_4 >= s_2^2.  The result is applied to show that

    (3, (1+\sqrt{17})/2, (1-\sqrt{17})/2, -2, -2) 

is not the spectrum of a nonnegative 5 x 5 matrix, while

    (3, (1+\sqrt{17})/2, (1-\sqrt{17})/2 ,-2, -2, 0) 

is the spectrum of a nonnegative symmetric 6 x 6 matrix.