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

Abstract

Let A be an entrywise nonnegative n n matrix and let s k := traceA k k = 1; 2; : : : . It is shown that if n is odd and s 1 = 0, then n , 1s 4 s 2 2 . The result is applied to show that 3; 1 2 1 p 17; ,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.