Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of General Matrices

Abstract

In the present paper, formulas for the Rayleigh-quotient representation of the real parts, imaginaryparts, and moduli of the eigenvalues of general matrices are obtained that resemble correspondingformulas for the eigenvalues of self-adjoint matrices. These formulas are of interest in Linear Algebraand in the theory of linear dynamical systems. The key point is that a weighted scalar product isused that is defined by means of a special positive definite matrix. As applications, one obtainsconvexity properties of newly-defined numerical ranges of a matrix. A numerical example underpinsthe theoretical findings.