Related bounds for the extreme eigenvalues

We show how to construct an increasing (decreasing) sequence of lower (upper) bounds for the smallest (largest) eigenvalue of a matrix with real spectrum by using the knowledge of the largest (smallest) eigenvalue. We apply these results to obtain a decreasing sequence of upper bounds for the largest singular value of a complex matrix. Finally, we show that the new bounds improve some previous results. Examples in which the bounds could be useful are given.