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

Year 2020, Volume: 3 Issue: 2, 55 - 75, 31.08.2020

Ludwig Kohaupt

https://doi.org/10.33187/jmsm.669216

Abstract

In the present paper, formulas for the Rayleigh-quotient representation of the real parts, imaginary
parts, and moduli of the eigenvalues of general matrices are obtained that resemble corresponding
formulas for the eigenvalues of self-adjoint matrices. These formulas are of interest in Linear Algebra
and in the theory of linear dynamical systems. 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

Supporting Institution

No supporting institution

Project Number

No project number

References

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