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

Dedicated to Hans Schneider on the occasion of his seventieth birthday. Abstract. Let A be an entrywise nonnegative n n matrix and let sk := trace (A) (k = 1;2; : : :). It is shown that if n is odd and s1 = 0, then (n 1)s4 s22. The result is applied to show that (3; 1 2 (1 p17); 2; 2) is not the spectrum of a nonnegative 5 5 matrix while (3; 1 2 (1 p 17); 2; 2;0) is the spectrum of a nonnegative symmetric 6 6 matrix.