LOWER BOUNDS FOR THE SPREAD OF A NONNEGATIVE MATRIX

Given an integer n >= 2 and a real number a >= 0, let C-n(a) be the collection of all nonnegative n x n matrices A = [a(i, j)](i, j=1)(n) such that a = min(1 <= i <= n) a(i, i) and r(A) > a, where r(A) denotes the spectral radius of A. We prove some lower bounds for the spread s(A) of A subset of C-n(a) that is defined as the maximum distance between any two eigenvalues of A. In particular, we prove that s(A) > 2/2+root 2n (r(A) - a) for all A is an element of C-n(a).