Intervals of certain classes of Z-matrices

• Volume: 34, Issue: 1, page 85-93
• ISSN: 1509-9415

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Abstract

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Let A and B be M-matrices satisfying A ≤ B and J = [A,B] be the set of all matrices C such that A ≤ C ≤ B, where the order is component wise. It is rather well known that if A is an M-matrix and B is an invertible M-matrix and A ≤ B, then aA + bB is an invertible M-matrix for all a,b > 0. In this article, we present an elementary proof of a stronger version of this result and study corresponding results for certain other classes as well.

How to cite

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M. Rajesh Kannan, and K.C. Sivakumar. "Intervals of certain classes of Z-matrices." Discussiones Mathematicae - General Algebra and Applications 34.1 (2014): 85-93. <http://eudml.org/doc/270177>.

@article{M2014,
abstract = {Let A and B be M-matrices satisfying A ≤ B and J = [A,B] be the set of all matrices C such that A ≤ C ≤ B, where the order is component wise. It is rather well known that if A is an M-matrix and B is an invertible M-matrix and A ≤ B, then aA + bB is an invertible M-matrix for all a,b > 0. In this article, we present an elementary proof of a stronger version of this result and study corresponding results for certain other classes as well.},
author = {M. Rajesh Kannan, K.C. Sivakumar},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {interval matrix; M-matrix; N-matrix; N₀-matrix; nonnegativity; interval hull of matrices; -matrix; Moore-Penrose inverse; -matrix; group inverse; range kernel regularity; range-symmetric matrix},
language = {eng},
number = {1},
pages = {85-93},
title = {Intervals of certain classes of Z-matrices},
url = {http://eudml.org/doc/270177},
volume = {34},
year = {2014},
}

TY - JOUR
AU - M. Rajesh Kannan
AU - K.C. Sivakumar
TI - Intervals of certain classes of Z-matrices
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2014
VL - 34
IS - 1
SP - 85
EP - 93
AB - Let A and B be M-matrices satisfying A ≤ B and J = [A,B] be the set of all matrices C such that A ≤ C ≤ B, where the order is component wise. It is rather well known that if A is an M-matrix and B is an invertible M-matrix and A ≤ B, then aA + bB is an invertible M-matrix for all a,b > 0. In this article, we present an elementary proof of a stronger version of this result and study corresponding results for certain other classes as well.
LA - eng
KW - interval matrix; M-matrix; N-matrix; N₀-matrix; nonnegativity; interval hull of matrices; -matrix; Moore-Penrose inverse; -matrix; group inverse; range kernel regularity; range-symmetric matrix
UR - http://eudml.org/doc/270177
ER -

References

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1. [1] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (SIAM, Philadelphia, 1994). Zbl0815.15016
2. [2] R.W. Cottle, A field guide to the matrix classes found in the literature of the linear complementarity problem, J. Global Optim. 46 (2010) 571-580. doi: 10.1007/s10898-009-9441-z Zbl1193.90203
3. [3] L. Hogben, Discrete Mathematics and Its Applications: Handbook of Linear Algebra (CRC Press, 2006).
4. [4] G.A. Johnson, A generalization of N-matrices, Linear Algebra Appl. 48 (1982) 201-217. doi: 10.1016/0024-3795(82)90108-2
5. [5] Ky Fan, Some matrix inequalities, Abh. Math. Sem. Univ. Hamburg 29 (1966) 185-196. doi: 10.1007/BF03016047
6. [6] A. Neumaier, Interval Methods for Systems of Equations, Encyclopedia of Mathematics and its Applications (Cambridge University Press, 1990).
7. [7] T. Parthasarathy and G. Ravindran, N-matrices, Linear Algebra Appl. 139 (1990) 89-102. doi: 10.1016/0024-3795(90)90390-X
8. [8] R. Smith and Shu-An Hu, Inequalities for monotonic pairs of Z-matrices, Lin. Mult. Alg. 44 (1998) 57-65. doi: 10.1080/03081089808818548 Zbl0907.15015
9. [9] R.S. Varga, Matrix Iterative Analysis, Springer Series in Computational Mathematics (Springer, New York, 2000). Zbl0998.65505

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