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On the Inverse of Block Tridiagonal Matrices with Applications to the Inverses of Band Matrices and Block Band Matrices

  • Pál Rózsa
  • Roberto Bevilacqua
  • Paola Favati
  • Francesco Romani
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 40)

Abstract

In the present paper the authors make an attempt to give a uniform description of the main properties of tridiagonal, band, block tridiagonal and block band matrices and their inverses. Some basic concepts are recalled and also some new results are presented.

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References

  1. 1.
    V. Amato, A method for the solution of difference equations, La scuola in Azione 13:109–116 (1962-63).Google Scholar
  2. 2.
    E. Asplund, Inverses of matrices {aij} which satisfy aij = 0 for j > i+p, Math. Scand. 7:57–60 (1959).Google Scholar
  3. 3.
    W.W. Barret, A theorem on inverses of tridiagonal matrices, Linear Algebra Appl. 27:211–217 (1979).CrossRefGoogle Scholar
  4. 4.
    W.W. Barret, P.J. Feinsilver, Inverses of band matrices, Linear Algebra Appl. 41:111–130 (1981).CrossRefGoogle Scholar
  5. 5.
    R. Bevilacqua, M. Capovani, Proprietà delle matrici tridiagonali ad elementi ed a blocchi, Editrice Tecnico Scientifica, Pisa, 1972.Google Scholar
  6. 6.
    R. Bevilacqua, M. Capovani, Proprietà delle matrici a banda ad elementi ed a blocchi, Boll. Un. Mat. Ital. B 13: 844–861 (1976).Google Scholar
  7. 7.
    R. Bevilacqua, B. Codenotti and F. Romani, Parallel solution of block tridiagonal linear systems, Linear Algebra Appl. 104:39–57 (1988).CrossRefGoogle Scholar
  8. 8.
    B. Bukhberger, G.A. Emel’yanenko, Methods of inverting tridiagonal matrices, Zh. Vychisl. Mat.i Fiz. 13:10–20 (1973).Google Scholar
  9. 9.
    W.L. Cao, W.J. Stewart, A note on Hessenberg-like Matrices, Linear Algebra Appl. 76:233–240 (1986).CrossRefGoogle Scholar
  10. 10.
    M. Capovani, Sulla determinazione della inversa delle matrici tridiagonali e tridiagonali a blocchi, Calcolo 7:295–303 (1970).CrossRefGoogle Scholar
  11. 11.
    M. Capovani, Su alcune proprietà delle matrici tridiagonali e pentadiagonali, Calcolo 8:149–159 (1971).CrossRefGoogle Scholar
  12. 12.
    F.R. Gantmacher, M.G. Krein, Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen Mechanischer Systeme, Akademie Verlag, Berlin, 1960.Google Scholar
  13. 13.
    S.B. Haley, Solution of band matrix equations by projection-recurrence, Linear Algebra Appl. 32:33–48 (1980).CrossRefGoogle Scholar
  14. 14.
    Y. Ikebe, On inverses of Hessenberg matrices, Linear Algebra Appl. 24:93–97 (1979).CrossRefGoogle Scholar
  15. 15.
    E.G. Kounias, An inversion technique for certain patterned matrices, J. Math. Anal. Appl. 21:695–698 (1968).CrossRefGoogle Scholar
  16. 16.
    J.W. Lewis, Inversion of tridiagonal matrices, Numer. Math. 38:333–345 (1982).CrossRefGoogle Scholar
  17. 17.
    E. Neuman, The inversion of certain band matrices, Rocz.Pol.Tow. Mat. Ser.3 9:15–24 (1977).Google Scholar
  18. 18.
    T. Oohashi, Some representation for inverses of band matrices, TRU Math. 14-2:39–47 (1978).Google Scholar
  19. 19.
    S.A.H. Rizvi, Inverses of quasi-tridiagonal matrices, Linear Algebra Appl. 56:177–184 (1984).CrossRefGoogle Scholar
  20. 20.
    F. Romani, On the additive structure of the inverses of banded matrices, Linear Algebra Appl. 80:131–140 (1986).CrossRefGoogle Scholar
  21. 21.
    P. Rózsa, Band matrices and semi-separable matrices, Colloq.Math. Soc. János Bolyai 50:229–237 (1986).Google Scholar
  22. 22.
    P. Rózsa, On the inverse of band matrices, Integral Equations Operator Theory 10:82–95 (1987).CrossRefGoogle Scholar
  23. 23.
    S. Schechter, Quasi-tridiagonal matrices and type-insensitive difference equations, Quart. Appl. Math. 18:285–295 (1960-61).Google Scholar
  24. 24.
    D. Szynal, J. Szynal, A propos de l’inversion des matrices généralisées de Jacobi, Apl. Mat. 17:28–32 (1972).Google Scholar
  25. 25.
    T. Torii, Inversion of tridiagonal matrices and the stability of tridiagonal systems of linear equations, Tech. Rep. Osaka Univ. 16:403–414 (1966).Google Scholar
  26. 26.
    V.R.R. Uppuluri, J.A. Carpenter, An inversion method for band matrices, J. Math. Anal. Appl. 31:554–558 (1970).CrossRefGoogle Scholar
  27. 27.
    T. Yamamoto, Y. Ikebe, Inversion of band matrices, Linear Algebra Appl. 24:105–111 (1979).CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel 1989

Authors and Affiliations

  • Pál Rózsa
    • 1
  • Roberto Bevilacqua
    • 2
  • Paola Favati
    • 3
  • Francesco Romani
    • 2
  1. 1.Department of MathematicsTechnical University of BudapestBudapestHungary
  2. 2.Dipartimento di InformaticaUniversity of PisaPisaItaly
  3. 3.Istituto di Elaborazione dell’InformazioneCNRPisaItaly

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