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Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of Simple Eigenvalues of Compact Operators

Yıl 2022, , 48 - 77, 30.06.2022

Ludwig Kohaupt

https://doi.org/10.33434/cams.1020515

Öz

In an earlier paper, the author derived generalized Rayleigh-quotient formulas for
the real parts, imaginary parts, and moduli of the eigenvalues of diagonalizable
matrices. More precisely, max-, min-max-, min-, and max-min-formulas were
obtained. In this paper, we extend these results to the eigenvalues of linear
nonsymmetric compact operators with simple eigenvalues in a Hilbert space. As
an application, a new formula for the spectral radius is derived. An example
arising from a boundary value problem in Mathematical Physics illustrates the
general results, and numerical computations underpin the theoretical findings.
In addition, the Euler column is treated from the area of Elastomechanics, which
is complemented by references to other examples from this area.

Anahtar Kelimeler

Generalized Rayleigh-Quotients Hilbert space Real parts Imaginary parts and moduli of eigenvalues Simple eigenvalues of compact operators

Destekleyen Kurum

There is no supporting institution

Kaynakça

  • [ 1] N.I. Achieser, I.M. Glasmann, Theorie der linearen Operatoren im Hilbert-Raum (Theory of Linear Operators in Hilbert Space; German Translation of the Russian Original), Akademie- Verlag, Berlin, 1968.
  • [ 2] W.E. Boyce, R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 8th Edition, John Wiley & Sons, Inc., Hoboke, 2005.
  • [ 3] E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, New York Toronto London, 1955.
  • [ 4] L. Collatz, Eigenwertaufgaben mit technischen Anwendungen (Eigenvalue Problems with Applications in Engineering), Geest & Portig K.-G., Leipzig, 1949.
  • [ 5] R. Courant, D. Hilbert, Mathematische Methoden der Physik, Band 1 (Methods of Mathematical Physics, Vol. 1), Springer-Verlag, Berlin, Heidelberg, New York, 1968.
  • [ 6] R.D. Grigorieff, Diskrete Approximation von Eigenwertproblemen. III: Asymptotische Entwicklungen (Discrete Approximation of Eigenvalue Problems. III: Asymptotic Expansions), Num. Math. 25(1975)79-97.
  • [ 7] H. Heuser, Funktionalanalysis (Functional Analysis), B.G. Teubner, Stuttgart, 1975.
  • [ 8] L.V. Kantorovich, G.P. Akilov, Functional Analysis in Normed Spaces; English Translation of the Russian Original), Pergamon Press, 1964.
  • [ 9] L.W.Kantorowitsch, G.P. Akilow, Funktionalanalysis in normierten R¨aumen (Functional Analysis in Normed Spaces; German Translation of the Russian Original), Akademie-Verlag Berlin, 1965.
  • [10] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
  • [11] K. Knopp, Theory of Functions, Parts I and II (English Translation of the German Original), Dover Publications, 1975.
  • [12] L. Kohaupt, Construction of a Biorthogonal System of Principal Vectors for Matrices A and A* with Applications to dx/dt˙ = Ax; x(t0) = x0, Journal of Computational Mathematics and Optimization 3(3)(2007)163-192.
  • [13] L. Kohaupt, Biorthogonalization of the Principal Vectors for the Matrices A and A* with Applications to the Computation of the Explicit Representation of the Solution x(t) of dy/dt = Ax; x(t0) = x0, Applied Mathematical Sciences 2(20)(2008)961-974.
  • [14] L. Kohaupt, Introduction to a Gram-Schmidt-type Biorthogonalization Method, Rocky Mountain Journal of Mathematics 44(4)(2014)1265-1279.
  • [15] L. Kohaupt, Generalized Rayleigh-quotient formulas for the eigenvalues of self-adjoint matrices, Journal of Algebra and Applied Mathematics 14(1)(2016)1-26.
  • [16] L. Kohaupt, Generalized Rayleigh-quotient formulas for the eigenvalues of diagonalizable matrices, Journal of Advances in Mathematics 14(2)(2018)7702-7728.
  • [17] L. Kohaupt, Generalized Rayleigh-quotient formulas for the eigenvalues of general matrices, Univeral Journal of Mathematics and Applications 4(1)(2021)9-25.
  • [18] L. Kohaupt, Eigenvalue Expansion of Nonsymmetric Linear Compact Operators in Hilbert Space, CAMS IV(2)(2021)55-74.
  • [19] W. Luther, K. Niederdrenk, F. Reutter, H. Yserentant, Gew¨ohnliche Differentialgleichungen, Analytische und numerische Behandlung (Ordinary Differential Equations, Analytic and Numerical Treatment), Vieweg, Braunschweig Wiesbaden, 1987.
  • [20] S.G. Michlin, Variationsmethoden der Mathematischen Physik (Variational Methods of Mathematical Physics; German Translation of the Russian Original), Akademie-Verlag, Berlin, 1962.
  • [21] B.N. Parlett, D.R. Taylor, Z.A.Liu, A look-ahead Lanczos algorithm for unsymmetric matrices, Mathematics of Computation 44(169)(1985)105-124.
  • [22] E.C. Pestel, F.A. Leckie, Matrix Methods in Elastomechanics, McGraw-Hill Book Company, Inc., New York San Francisco Toronto London, 1963.
  • [23] W. Schnell, D. Gross, W. Hauger, Technische Mechanik, Band 2: Elastostatik (Mechanics for Engineers, Vol. 2: Elastostatics), Springer-Verlag, Berlin Heidelberg New York Tokyo, 1985.
  • [24] F. Stummel, Diskrete Konvergenz linearer Operatoren II (Discrete Convergence of Linear Operators, Part II), Mathematische Zeitschrift 120 (1971)231-264.
  • [25] F. Stummel, K. Hainer, Introduction to Numerical Analysis, Scottish Academic Press, Edinburgh, 1980.
  • [26] F. Stummel, L. Kohaupt, Eigenwertaufgaben in Hilbertschen R¨aumen. Mit Aufgaben und vollst¨andigen L¨osungen (Eigenvalue Problems in Hilbert Spaces. With Exercises and Complete Solutions), Logos Verlag, Berlin, 2021.
  • [27] A.E. Taylor, Introduction to Functional Analysis, John Wiley & Sons, New York London, 1958.
  • [28] St. P. Timoshenko, J. M. Gere, Theory of Elastic Stability, McGraw-Hill Kogakusha, Ltd., Tokyo et al., 1961.
  • [29] W. Walter, Gew¨ohnliche Differentialgleichungen. Eine Einf¨uhrung. (Ordinary Differential Equations. An Introduction), Springer, Berlin et al., 2000.
  • [30] D. Werner, Funktionalanalysis (Functional Analysis), 8th Edition, Springer Spektrum, 2018.

Yıl 2022, , 48 - 77, 30.06.2022

Ludwig Kohaupt

https://doi.org/10.33434/cams.1020515

Öz

Kaynakça

  • [ 1] N.I. Achieser, I.M. Glasmann, Theorie der linearen Operatoren im Hilbert-Raum (Theory of Linear Operators in Hilbert Space; German Translation of the Russian Original), Akademie- Verlag, Berlin, 1968.
  • [ 2] W.E. Boyce, R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 8th Edition, John Wiley & Sons, Inc., Hoboke, 2005.
  • [ 3] E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, New York Toronto London, 1955.
  • [ 4] L. Collatz, Eigenwertaufgaben mit technischen Anwendungen (Eigenvalue Problems with Applications in Engineering), Geest & Portig K.-G., Leipzig, 1949.
  • [ 5] R. Courant, D. Hilbert, Mathematische Methoden der Physik, Band 1 (Methods of Mathematical Physics, Vol. 1), Springer-Verlag, Berlin, Heidelberg, New York, 1968.
  • [ 6] R.D. Grigorieff, Diskrete Approximation von Eigenwertproblemen. III: Asymptotische Entwicklungen (Discrete Approximation of Eigenvalue Problems. III: Asymptotic Expansions), Num. Math. 25(1975)79-97.
  • [ 7] H. Heuser, Funktionalanalysis (Functional Analysis), B.G. Teubner, Stuttgart, 1975.
  • [ 8] L.V. Kantorovich, G.P. Akilov, Functional Analysis in Normed Spaces; English Translation of the Russian Original), Pergamon Press, 1964.
  • [ 9] L.W.Kantorowitsch, G.P. Akilow, Funktionalanalysis in normierten R¨aumen (Functional Analysis in Normed Spaces; German Translation of the Russian Original), Akademie-Verlag Berlin, 1965.
  • [10] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
  • [11] K. Knopp, Theory of Functions, Parts I and II (English Translation of the German Original), Dover Publications, 1975.
  • [12] L. Kohaupt, Construction of a Biorthogonal System of Principal Vectors for Matrices A and A* with Applications to dx/dt˙ = Ax; x(t0) = x0, Journal of Computational Mathematics and Optimization 3(3)(2007)163-192.
  • [13] L. Kohaupt, Biorthogonalization of the Principal Vectors for the Matrices A and A* with Applications to the Computation of the Explicit Representation of the Solution x(t) of dy/dt = Ax; x(t0) = x0, Applied Mathematical Sciences 2(20)(2008)961-974.
  • [14] L. Kohaupt, Introduction to a Gram-Schmidt-type Biorthogonalization Method, Rocky Mountain Journal of Mathematics 44(4)(2014)1265-1279.
  • [15] L. Kohaupt, Generalized Rayleigh-quotient formulas for the eigenvalues of self-adjoint matrices, Journal of Algebra and Applied Mathematics 14(1)(2016)1-26.
  • [16] L. Kohaupt, Generalized Rayleigh-quotient formulas for the eigenvalues of diagonalizable matrices, Journal of Advances in Mathematics 14(2)(2018)7702-7728.
  • [17] L. Kohaupt, Generalized Rayleigh-quotient formulas for the eigenvalues of general matrices, Univeral Journal of Mathematics and Applications 4(1)(2021)9-25.
  • [18] L. Kohaupt, Eigenvalue Expansion of Nonsymmetric Linear Compact Operators in Hilbert Space, CAMS IV(2)(2021)55-74.
  • [19] W. Luther, K. Niederdrenk, F. Reutter, H. Yserentant, Gew¨ohnliche Differentialgleichungen, Analytische und numerische Behandlung (Ordinary Differential Equations, Analytic and Numerical Treatment), Vieweg, Braunschweig Wiesbaden, 1987.
  • [20] S.G. Michlin, Variationsmethoden der Mathematischen Physik (Variational Methods of Mathematical Physics; German Translation of the Russian Original), Akademie-Verlag, Berlin, 1962.
  • [21] B.N. Parlett, D.R. Taylor, Z.A.Liu, A look-ahead Lanczos algorithm for unsymmetric matrices, Mathematics of Computation 44(169)(1985)105-124.
  • [22] E.C. Pestel, F.A. Leckie, Matrix Methods in Elastomechanics, McGraw-Hill Book Company, Inc., New York San Francisco Toronto London, 1963.
  • [23] W. Schnell, D. Gross, W. Hauger, Technische Mechanik, Band 2: Elastostatik (Mechanics for Engineers, Vol. 2: Elastostatics), Springer-Verlag, Berlin Heidelberg New York Tokyo, 1985.
  • [24] F. Stummel, Diskrete Konvergenz linearer Operatoren II (Discrete Convergence of Linear Operators, Part II), Mathematische Zeitschrift 120 (1971)231-264.
  • [25] F. Stummel, K. Hainer, Introduction to Numerical Analysis, Scottish Academic Press, Edinburgh, 1980.
  • [26] F. Stummel, L. Kohaupt, Eigenwertaufgaben in Hilbertschen R¨aumen. Mit Aufgaben und vollst¨andigen L¨osungen (Eigenvalue Problems in Hilbert Spaces. With Exercises and Complete Solutions), Logos Verlag, Berlin, 2021.
  • [27] A.E. Taylor, Introduction to Functional Analysis, John Wiley & Sons, New York London, 1958.
  • [28] St. P. Timoshenko, J. M. Gere, Theory of Elastic Stability, McGraw-Hill Kogakusha, Ltd., Tokyo et al., 1961.
  • [29] W. Walter, Gew¨ohnliche Differentialgleichungen. Eine Einf¨uhrung. (Ordinary Differential Equations. An Introduction), Springer, Berlin et al., 2000.
  • [30] D. Werner, Funktionalanalysis (Functional Analysis), 8th Edition, Springer Spektrum, 2018.
Toplam 30 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Ludwig Kohaupt

Yayımlanma Tarihi 30 Haziran 2022
Gönderilme Tarihi 8 Kasım 2021
Kabul Tarihi 28 Nisan 2022
Yayımlandığı Sayı Yıl 2022

Kaynak Göster

APA Kohaupt, L. (2022). Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of Simple Eigenvalues of Compact Operators. Communications in Advanced Mathematical Sciences, 5(2), 48-77. https://doi.org/10.33434/cams.1020515
AMA Kohaupt L. Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of Simple Eigenvalues of Compact Operators. Communications in Advanced Mathematical Sciences. Haziran 2022;5(2):48-77. doi:10.33434/cams.1020515
Chicago Kohaupt, Ludwig. “Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of Simple Eigenvalues of Compact Operators”. Communications in Advanced Mathematical Sciences 5, sy. 2 (Haziran 2022): 48-77. https://doi.org/10.33434/cams.1020515.
EndNote Kohaupt L (01 Haziran 2022) Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of Simple Eigenvalues of Compact Operators. Communications in Advanced Mathematical Sciences 5 2 48–77.
IEEE L. Kohaupt, “Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of Simple Eigenvalues of Compact Operators”, Communications in Advanced Mathematical Sciences, c. 5, sy. 2, ss. 48–77, 2022, doi: 10.33434/cams.1020515.
ISNAD Kohaupt, Ludwig. “Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of Simple Eigenvalues of Compact Operators”. Communications in Advanced Mathematical Sciences 5/2 (Haziran 2022), 48-77. https://doi.org/10.33434/cams.1020515.
JAMA Kohaupt L. Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of Simple Eigenvalues of Compact Operators. Communications in Advanced Mathematical Sciences. 2022;5:48–77.
MLA Kohaupt, Ludwig. “Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of Simple Eigenvalues of Compact Operators”. Communications in Advanced Mathematical Sciences, c. 5, sy. 2, 2022, ss. 48-77, doi:10.33434/cams.1020515.
Vancouver Kohaupt L. Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of Simple Eigenvalues of Compact Operators. Communications in Advanced Mathematical Sciences. 2022;5(2):48-77.

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