- Journal of Mathematical Sciences and Modelling
- Vol: 3 Issue: 2
- Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues o...
Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of General Matrices
Authors : Ludwig Kohaupt
Pages : 55-75
View : 10 | Download : 3
Publication Date : 2020-08-31
Article Type : Research
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