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