Ludwig KOHAUPT

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

In the present paper, formulas for the Rayleigh-quotient representation of the real parts, imaginary parts, and moduli of the eigenvalues of diagonalizable matrices are obtained that resemble corresponding formulas for the eigenvalues of self-adjoint matrices. These formulas are new and of interest in Linear Algebra and in the theory of linear dynamical systems. Since the style of paper is expository, it could also be of interest in graduate/undergraduate teaching or research at college level. 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 Weighted norm,

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[1] 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.
[2] L. Kohaupt, Construction of a biorthogonal system of principal vectors of the matrices A and A with applications to the initial value problem x = Ax; x(t0) = x0, J. Comp. Math. Opt. 3(3)(2007)163-192.
[3] 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 = Ax; x(t0) = x0, Appl. MAth. Sci. 2(20)(2008)961-974.
[4] L. Kohaupt, Solution of the vibration problem M¨ y+B˙ y+Ky = 0; y(t0) = y0; ˙ y(t0) = ˙ y0 without the hypothesis BM?1K = KM?1B or B = aM+bK, Appl. MAth. Sci. 2(41)(2008)1989-2024.
[5] L. Kohaupt, Solution of the matrix eigenvalue problem VA+AV = mV 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 C1B with positive definite matrix C and Hermitian B as well as applications, J. Appl. Math. Comput., DOI 10.1007/s12190-015-0876-8, (2015) 28 pages.
[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] F. Stummel, K. Hainer, Introduction to Numerical Analysis, Scottish Academic Press, Edinburgh, 1980.