Estimation of the eigenvalues and the smallest singular value of matrices

The purpose of this paper is to discuss the estimation of eigenvalues and the smallest singular value of matrices. Firstly, we prove that all the eigenvalues of arbitrarily complex matrix are located in one closed disk around trMn of radiusn-1n‖M‖F2-|trM|2n-max1⩽k⩽n-1‖Bk×(n-k)‖F-‖C(n-k)×k‖F2.Secondly, we present a lower bound for the smallest singular value of matrices. Finally, we give some numerical examples which will show the effectiveness of our results.