New eigenvalue estimates for complex matrices (Q2365932)

Let \(A\) be an \(n\times n\) complex matrix with eigenvalues \(\lambda_ 1,\ldots,\lambda_ n\). The authors prove that all of the eigenvalues lie in the rectangle \[ [({\mathfrak R}(\text{tr }A)/n)-\alpha,({\mathfrak R}(\text{tr }A)/n)+\alpha]\times[({\mathfrak I}(\text{tr }A)/n)-\beta,({\mathfrak I}(\text{tr }A)/n)+\beta], \] where \(\alpha=\left[(n-1)/n(\sum^ n_{k=1}({\mathfrak R}\lambda_ k)^ 2-({\mathfrak R}(\text{tr }A))^ 2/n)\right]^{1/2}\), and \(\beta=\left[(n-1)/n(\sum^ n_{k=1}({\mathfrak I}\lambda_ k)^ 2- ({\mathfrak I}(\text{tr }A))^ 2/n)\right]^{1/2}\). Also, some bounds for the rectangle which can be computed without knowing the eigenvalues are given.