Generalized Rayleigh-quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of Diagonalizable Matrices
In the present paper, generalized Rayleigh-quotient formulas for the real parts, imaginary parts, and moduli of the eigenvalues of diagonalizable matrices are derived. These formulas are new and correspond to similar formulas for the eigenvalues of self-adjoint matrices obtained recently. Numerical examples underpin the theoretical findings.
R.T. Gregory, D.L. Karney, A Collection of Matrices for Testing Computational Algorithms, Wiley-Interscience, New York London Sydney Toronto, 1969.
R.A. Horn, Ch.R. Johnson, Matrix Analysis, University Press, Cambridge, 22nd printing, 2009.
R.A. Horn, Ch.R. Johnson, Topics in Matrix Analysis, University Press, Cambridge, 10th printing 2008.
T. Kato, Perturbation Theory for Linear Operators, Springer, New York, 1966.
L. Kohaupt, Construction of a biorthogonal system of principal vectors of the matrices A and A* with applications to the initial value problem x = A x, x(t0) = x0, Journal of Computational Mathematics and Optimization 3(3)(2007)163-192.
L. Kohaupt, Biorthogonalization of the principal vectors for the matrices A and A* with application to the computation of the explicit representation of the solution x(t) of x = A x, x(t0) = x0, Applied Mathematical Sciences 2(20)(2008)961-974.
L. Kohaupt, Generalized Rayleigh-quotient representation of the eigenvalues of self-adjoint matrices, Journal of Algebra and Applied Mathematics 14(1)(2016)1-26.
L. Kohaupt, Rayleigh-quotient representation of the real parts, imaginary parts, and moduli of the eigenvalues of diagonalizable matrices, to be submitted for publication.
F. Stummel, K. Hainer, Introduction to Numerical Analysis, Scottish Academic Press, Edinburgh, 1980.
L.H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.
R. Zurm¨uhl, S. Falk, Matrizen und ihre Anwendungen, Teil 1: Grundlagen (Matrices and Their Applications, Part 1: Foundations), Springer-Verlag, Berlin Heidelberg New York Tokyo, 1984.
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