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Ultrametric Matrices

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2118)

Abstract

This chapter is devoted to the study of ultrametric matrices introduced by Martínez, Michon and San Martín in [44], where it was proved that the inverse of an ultrametric matrix is a row diagonally dominant Stieltjes matrix (a particular case of an M-matrix). We shall include this result in Theorem 3.5 and give a proof in the lines done by Nabben and Varga in [51]. One of the important aspects of ultrametric matrices is that they represent a class of inverse M-matrices described in very simple combinatorial terms.

Keywords

  • Tree Matrice
  • Geodesic Distance
  • Oriented Graph
  • Ultrametric Space
  • Ternary Relation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Fig. 3.1
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References

  1. J.P. Benzecri et collaborateurs, LÁnalyse des données (Dunod, Paris, 1973)

    Google Scholar 

  2. D. Capocacia, M. Cassandro, P. Picco, On the existence of ther- modynamics for the generalized random energy model. J. Stat. Phys. 46(3/4), 493–505 (1987)

    CrossRef  Google Scholar 

  3. K.L. Chung, Markov Chains with Stationary Transition Probabilities (Springer, New York, 1960)

    CrossRef  MATH  Google Scholar 

  4. P. Dartnell, S. Martínez, J. San Martín, Opérateurs filtrés et chaînes de tribus invariantes sur un espace probabilisé dénombrable. Séminaire de Probabilités XXII Lecture Notes in Mathematics, vol. 1321 (Springer, New York, 1988)

    Google Scholar 

  5. C. Dellacherie, Private Communication (1985)

    Google Scholar 

  6. L.R. Ford, D.R. Fulkerson, Flows in Networks (Princeton University Press, Princeton, 1973)

    Google Scholar 

  7. R.E. Gomory, T. C. Hu, Multi-terminal network flows. SIAM J. Comput. 9(4), 551–570 (1961)

    MATH  MathSciNet  Google Scholar 

  8. R. Horn, C. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985)

    CrossRef  MATH  Google Scholar 

  9. T. Markham, Nonnegative matrices whose inverse are M-matrices. Proc. AMS 36, 326–330 (1972)

    MathSciNet  Google Scholar 

  10. S. Martínez, G. Michon, J. San Martín, Inverses of ultrametric matrices are of Stieltjes types. SIAM J. Matrix Anal. Appl. 15, 98–106 (1994)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. S. Martínez, J. San Martín, X. Zhang, A new class of inverse M-matrices of tree-like type. SIAM J. Matrix Anal. Appl. 24(4), 1136–1148 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. J.J. McDonald, M. Neumann, H. Schneider, M.J. Tsatsomeros. Inverse M-matrix inequalities and generalized ultrametric matrices. Linear Algebra Appl. 220, 321–341 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. R. Nabben, R.S. Varga, A linear algebra proof that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant Stieltjes matrix. SIAM J. Matrix Anal. Appl. 15, 107–113 (1994)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. R. Nabben, R.S. Varga, Generalized ultrametric matrices – a class of inverse M-matrices. Linear Algebra Appl. 220, 365–390 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. R. Nabben, R. Varga, On classes of inverse Z-matrices. Linear Algebra Appl. 223/224, 521–552 (1998)

    Google Scholar 

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Dellacherie, C., Martinez, S., San Martin, J. (2014). Ultrametric Matrices. In: Inverse M-Matrices and Ultrametric Matrices. Lecture Notes in Mathematics, vol 2118. Springer, Cham. https://doi.org/10.1007/978-3-319-10298-6_3

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