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

Abstract

In the present paper, formulas for the Rayleigh-quotient representation of the real parts, imaginary parts, and moduli of the eigenvalues of diagonalizable matrices are obtained that resemble corresponding formulas for the eigenvalues of self-adjoint matrices. These formulas are new and of interest in Linear Algebra and in the theory of linear dynamical systems. Since the style of paper is expository, it could also be of interest in graduate/undergraduate teaching or research at college level. The key point is that a weighted scalar product is used that is defined by means of a special positive definite matrix. As applications, one obtains convexity properties of newly-defined numerical ranges of a matrix. A numerical example underpins the theoretical findings.

Keywords

• rayleigh quotient
• real and imaginary parts of eigenvalues
• moduli of eigenvalues
• asymptotic stability of dynamical systems
• circular damped eigenfrequencies
• weighted norm