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

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

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.

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  • [ 1] F. Stummel, K. Hainer, Introduction to Numerical Analysis, Scottish Academic Press, Edinburgh, 1980.
  • [ 2] L. Kohaupt, Rayleigh-quotient representation of the real parts, imaginary parts, and moduli of the eigenvalues of diagonalizable matrices, J. Math. Sci. Model., 2(2) (2019), 82-98.
  • [ 3] L. Kohaupt, Solution of the vibration problem $M\ddot{y}+B\dot{y}+K y = 0, \, y(t_0)=y_0, \, \dot{y}(t_0)=\dot{y}_0$ {\em without} the hypothesis $B M^{-1} K = K M^{-1} B$ or $B = \alpha M + \beta K$}, Appl. Math. Sci., 2(41) (2008), 1989-2024.
  • [ 4] A.Czornik, P. Jurga´s, Some properties of the spectral radius of a set of matrices, Int. J. Appl. Math. Sci., 16(2)(2006)183-188.
  • [ 5] L. Kohaupt, Solution of the matrix eigenvalue problem $V A + A^{\ast} V = \mu V$ with applications to the study of free linear systems, J. Comp. Appl. Math., 213(1) (2008), 142-165.
  • [ 6] L. Kohaupt, Spectral properties of the matrix $C^{-1} B$ with positive definite matrix $C$ and Hermitian $B$ as well as applications, J. Appl. Math. Comput., 50 (2016), 389-416.
  • [ 7] T.J. Laffey, H. ˘Smigoc, Nonnegatively realizable spectra with two positive eigenvalues, Linear Multilinear Algebra, 58(7-8) (2010), 1053-1069.
  • [ 8] P. Lancaster, Theory of Matrices, Academic Press, New York and London, 1969.
  • [ 9] P.C. M¨uller, W.O. Schiehlen, Linear Vibrations, Martinus Nijhoff Publishers, Dordrecht Boston Lancaster, 1985.
  • [10] S.V. Savchenko, On the change in the spectral properties of a matrix under perturbations of sufficiently low rank, Funct. Anal. Appl., 38(1) (2004), 69-71.
  • [11] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer, New York Heidelberg, Third Edition, 2010.
  • [12] L. Kohaupt, Construction of a biorthogonal system of principal vectors of the matrices $A$ and $A^{\ast}$ with applications to the initial value problem $\dot{x}=A\,x, \; x(t_0)=x_0$, J. Comput. Math. Optim., 3(3) (2007), 163-192.
  • [13] L. Kohaupt, Further spectral properties of the matrix$C^{-1} B$ with positive definite $C$ and Hermitian $B$ applied to wider classes of matrices $C$ and $B$, J. Appl. Math. Comput., 52 (2016), 215-243.
  • [14] L. Kohaupt, Biorthogonalization of the principal vectors for the matrices $A$ and $A^{\ast}$ with application to the computation of the explicit representation of the solution $x(t)$ of $\dot{x}=A\,x, \; x(t_0)=x_0$, Appl. Math. Sci., 2(20) (2008), 961-974.