Sharpening of the inequalities of Schur, Eberlein, Kress and Huang, and new location of eigenvalues (Q2928365)

Let \(A=a_{ij}\) be an \(\;n\times n\) \ complex matrix with eigenvalues \(\lambda _{1},\lambda _{2},...,\lambda _{n}\) . The author obtains a tighter upper bound for \(\sum\limits_{j=1}^{n}\left| \lambda _{j}\right| ^{2}\) than previous upper bounds given by Schur, Eberlein, Kress and Huang. The author derives five new determinantal inequalities. He also obtains methods to locate eigenvalues of a given matrix . Numerical examples are considered.