Research Article
BibTex RIS Cite
Year 2019, Volume: 2 Issue: 2, 82 - 98, 30.08.2019
https://doi.org/10.33187/jmsm.488921

Abstract

References

  • [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.

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

Year 2019, Volume: 2 Issue: 2, 82 - 98, 30.08.2019
https://doi.org/10.33187/jmsm.488921

Abstract

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.

References

  • [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.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ludwig Kohaupt 0000-0003-1781-0600

Publication Date August 30, 2019
Submission Date November 28, 2018
Acceptance Date January 17, 2019
Published in Issue Year 2019 Volume: 2 Issue: 2

Cite

APA Kohaupt, L. (2019). Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of Diagonalizable Matrices. Journal of Mathematical Sciences and Modelling, 2(2), 82-98. https://doi.org/10.33187/jmsm.488921
AMA Kohaupt L. Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of Diagonalizable Matrices. Journal of Mathematical Sciences and Modelling. August 2019;2(2):82-98. doi:10.33187/jmsm.488921
Chicago Kohaupt, Ludwig. “Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of Diagonalizable Matrices”. Journal of Mathematical Sciences and Modelling 2, no. 2 (August 2019): 82-98. https://doi.org/10.33187/jmsm.488921.
EndNote Kohaupt L (August 1, 2019) Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of Diagonalizable Matrices. Journal of Mathematical Sciences and Modelling 2 2 82–98.
IEEE L. Kohaupt, “Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of Diagonalizable Matrices”, Journal of Mathematical Sciences and Modelling, vol. 2, no. 2, pp. 82–98, 2019, doi: 10.33187/jmsm.488921.
ISNAD Kohaupt, Ludwig. “Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of Diagonalizable Matrices”. Journal of Mathematical Sciences and Modelling 2/2 (August 2019), 82-98. https://doi.org/10.33187/jmsm.488921.
JAMA Kohaupt L. Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of Diagonalizable Matrices. Journal of Mathematical Sciences and Modelling. 2019;2:82–98.
MLA Kohaupt, Ludwig. “Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of Diagonalizable Matrices”. Journal of Mathematical Sciences and Modelling, vol. 2, no. 2, 2019, pp. 82-98, doi:10.33187/jmsm.488921.
Vancouver Kohaupt L. Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of Diagonalizable Matrices. Journal of Mathematical Sciences and Modelling. 2019;2(2):82-98.

29237    Journal of Mathematical Sciences and Modelling 29238

                   29233

Creative Commons License The published articles in JMSM are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.