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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J.P. Benzecri et collaborateurs, LÁnalyse des données (Dunod, Paris, 1973)
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)
K.L. Chung, Markov Chains with Stationary Transition Probabilities (Springer, New York, 1960)
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)
C. Dellacherie, Private Communication (1985)
L.R. Ford, D.R. Fulkerson, Flows in Networks (Princeton University Press, Princeton, 1973)
R.E. Gomory, T. C. Hu, Multi-terminal network flows. SIAM J. Comput. 9(4), 551–570 (1961)
R. Horn, C. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985)
T. Markham, Nonnegative matrices whose inverse are M-matrices. Proc. AMS 36, 326–330 (1972)
S. Martínez, G. Michon, J. San Martín, Inverses of ultrametric matrices are of Stieltjes types. SIAM J. Matrix Anal. Appl. 15, 98–106 (1994)
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)
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)
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)
R. Nabben, R.S. Varga, Generalized ultrametric matrices – a class of inverse M-matrices. Linear Algebra Appl. 220, 365–390 (1995)
R. Nabben, R. Varga, On classes of inverse Z-matrices. Linear Algebra Appl. 223/224, 521–552 (1998)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-10298-6_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10297-9
Online ISBN: 978-3-319-10298-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)