Inverse Nonnegativity of Tridiagonal M-Matrices under Diagonal Element-Wise Perturbation


One of the most important properties of M-matrices is element-wise non-negative of its inverse. In this paper, we consider element-wise perturbations of tridiagonal M-matrices and obtain bounds on the perturbations so that the non-negative inverse persists. The largest interval is given by which the diagonal entries of the inverse of tridiagonal M-matrices can be perturbed without losing the property of total nonnegativity. A numerical example is given to illustrate our findings.

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Ramadan, M. and Abu Murad, M. (2015) Inverse Nonnegativity of Tridiagonal M-Matrices under Diagonal Element-Wise Perturbation. Advances in Linear Algebra & Matrix Theory, 5, 37-45. doi: 10.4236/alamt.2015.52004.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] McDonald, J.J., Nabben, R., Neumannand, M., Schneider, H. and Tsatsomeros, M.J. (1998) Inverse Tridiagonal Z-Matrices. Linear and Multilinear Algebra, 45, 75-97.
[2] Berman, A. and Plemmons, R. (1979) Nonnegative Matrices in the Mathenlatical Sciences. Academic, New York.
[3] Pena, J.M. (1995) M-Matrices Whose Inverses Are Totally Positive. Linear Algebra and Its Applications, 221, 189-193.
[4] Ostrowski, A. (1937) über die Determinanten mit überwiegender Hauptdiagonale. Commentarii Mathematici Helvetici, 10, 69-96.
[5] Minkowski, H. (1900) Zur Theorie der Einheiten in den algebraischen Zahlkorper. Nachrichten von der Konigl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universitat zu Gottingen Mathematisch-Physikalische Klasse: Fachgruppe II, Nachrichten aus der Physik, Astronomie, Geophysik, Technik, 90-93.
[6] Minkowski, H. (1907) Diophantische Approximationen. Tuebner, Leipzig.
[7] Horn, R. and Johnson, C. (1991) Topics in Matrix Analysis. Cambridge University Press, Cambridge.
[8] Adm, M. and Garloff, J. (2014) Invariance of Total Nonnegativity of a Tridiagonal Matrix under Element-Wise Perturbation. Operators and Matrices, 8, 129-137.
[9] Ando, T. (1987) Totally Positive Matrices. Linear Algebra and Its Applications, 90, 165-219.
[10] Pinkus, A. (2010) Totally Positive Matrices. Cambridge Tracts in Mathematics (No. 181). Cambridge University Press, Cambridge.
[11] Fallat, S.M. and Johnson, C.R. (2011) Totally Nonnegative Matrices. Princeton University Press, Princeton, Oxford.
[12] Adam, M. and Garloff, J. (2013) Interval of Totally Nonnegative Matrices. Linear Algebra and Its Applications, 439, 3796-3806.

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