Improving bounds for eigenvalues of complex matrices using traces

Let A be a complex matrix of order n with eigenvalues and m be any integer satisfying . The bound for ∑|λj|2 by Kress, de Vries, and Wegmann is strengthened. Furthermore, new bounds are presented to estimate the spectral radius of A using m and traces of A, A2, 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 , , , and m.