Lower bounds for the Perron root of a nonnegative matrix

AbstractFor an n × n matrix A = (aij) ⩾ 0, we prove the Perron root r(A) satisfies the inequality r(A) ⩾ ∏1⩽i<j⩽n(aijaji)xixj∏i=1n (aii+q)xi2∑i=1nxi2−q for any x=(x1,…,xn)T⩾0 with ∑i=1nxi=1 and for any q⩾–min{aii}. Using this, a necessary condition for a real n × n matrix A with aii>0 and aij⩽0, i≠j, i, j = 1,…,n, to be an M-matrix is presented, viz. ∏1⩽i<j⩽naijajiaiiajjxixj < (1+q)q−∑i=1nxi2 ∑i=1nxi2 for any x=(x1,…,xn)T⩾0 with ∑i=1nxi=1 and for any q>0