Improving bounds for eigenvalues of complex matrices using traces

Let A be a complex matrix of order n with eigenvalues lambda(j) (j = 1, 2,..., n) and m be any integer satisfying rank A <= m <= n. The bound for Sigma vertical bar lambda(j)vertical bar (2) by Kress, de Vries, and Wegmann is strengthened. Furthermore, new bounds are presented to estimate the spectral radius of A using tit and traces of A, A(2), A*A and A*A - AA*. We also improve some Wolkowicz-Styan bounds and previous localization of eigenvalues in rectangular or elliptic regions using traces. Several simple lower bounds for the spectral radius are given, involving tr A, tr A(2), tr A(3), and m. (C) 2007 Elsevier Inc. All rights reserved.