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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