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

Yıl 2020, , 55 - 75, 31.08.2020

Ludwig Kohaupt

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

Öz

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.

Anahtar Kelimeler

Rayleigh quotient, Real and imaginary parts of eigenvalues, Moduli of eigenvalues, Asymptotic stability of dynamical systems, Circular damped eigenfrequencies

Destekleyen Kurum

No supporting institution

Proje Numarası

No project number

Kaynakça

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