Lower bounds for the numerical radius

Let A be a complex n x n matrix. We find lower bounds for its numerical radius r(A) = max {vertical bar x*Ax vertical bar vertical bar x is an element of C-n, x*x = 1}. First, we choose x satisfying x*x = 1 and compute vertical bar x*Ax vertical bar. We also improve the simple bound so obtained. Second, we apply r(A) = max {lambda (zA + zA*/2 )vertical bar z is an element of C, vertical bar z vertical bar = 1} where lambda denotes the largest eigenvalue. (c) 2005 Elsevier Inc. All rights reserved.