Inverse Nonnegativity of Tridiagonal M-Matrices under Diagonal Element-Wise Perturbation
Affiliation(s)
1 Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El-Kom, Egypt. 2 Department of Mathematics, Faculty of Science, Zagazig University, Ash Sharqiyah, Egypt.
1 Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El-Kom, Egypt. 2 Department of Mathematics, Faculty of Science, Zagazig University, Ash Sharqiyah, Egypt.
ABSTRACT
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.
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.
KEYWORDS
Totally Positive Matrix, Totally Nonnegative Matrix, Tridiagonal Matrices, Compound Matrix, Element-Wise Perturbations
Totally Positive Matrix, Totally Nonnegative Matrix, Tridiagonal Matrices, Compound Matrix, Element-Wise Perturbations
Cite this paper
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.
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.
References
[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.
http://dx.doi.org/10.1080/03081089808818578 [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.
http://dx.doi.org/10.1016/0024-3795(93)00244-T [4] Ostrowski, A. (1937) über die Determinanten mit überwiegender Hauptdiagonale. Commentarii Mathematici Helvetici, 10, 69-96.
http://dx.doi.org/10.1007/BF01214284 [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.
http://dx.doi.org/10.1007/978-3-663-16055-7 [7] Horn, R. and Johnson, C. (1991) Topics in Matrix Analysis. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511840371 [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.
http://dx.doi.org/10.7153/oam-08-06 [9] Ando, T. (1987) Totally Positive Matrices. Linear Algebra and Its Applications, 90, 165-219.
http://dx.doi.org/10.1016/0024-3795(87)90313-2 [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.
http://dx.doi.org/10.1515/9781400839018 [12] Adam, M. and Garloff, J. (2013) Interval of Totally Nonnegative Matrices. Linear Algebra and Its Applications, 439, 3796-3806.
http://dx.doi.org/10.1016/j.laa.2013.10.021
[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.
http://dx.doi.org/10.1080/03081089808818578 [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.
http://dx.doi.org/10.1016/0024-3795(93)00244-T [4] Ostrowski, A. (1937) über die Determinanten mit überwiegender Hauptdiagonale. Commentarii Mathematici Helvetici, 10, 69-96.
http://dx.doi.org/10.1007/BF01214284 [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.
http://dx.doi.org/10.1007/978-3-663-16055-7 [7] Horn, R. and Johnson, C. (1991) Topics in Matrix Analysis. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511840371 [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.
http://dx.doi.org/10.7153/oam-08-06 [9] Ando, T. (1987) Totally Positive Matrices. Linear Algebra and Its Applications, 90, 165-219.
http://dx.doi.org/10.1016/0024-3795(87)90313-2 [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.
http://dx.doi.org/10.1515/9781400839018 [12] Adam, M. and Garloff, J. (2013) Interval of Totally Nonnegative Matrices. Linear Algebra and Its Applications, 439, 3796-3806.
http://dx.doi.org/10.1016/j.laa.2013.10.021