# Intervals of certain classes of Z-matrices

M. Rajesh Kannan; K.C. Sivakumar

Discussiones Mathematicae - General Algebra and Applications (2014)

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

## Access Full Article

top## Abstract

top## How to cite

topM. 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

top- [1] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (SIAM, Philadelphia, 1994). Zbl0815.15016
- [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] L. Hogben, Discrete Mathematics and Its Applications: Handbook of Linear Algebra (CRC Press, 2006).
- [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] Ky Fan, Some matrix inequalities, Abh. Math. Sem. Univ. Hamburg 29 (1966) 185-196. doi: 10.1007/BF03016047
- [6] A. Neumaier, Interval Methods for Systems of Equations, Encyclopedia of Mathematics and its Applications (Cambridge University Press, 1990).
- [7] T. Parthasarathy and G. Ravindran, N-matrices, Linear Algebra Appl. 139 (1990) 89-102. doi: 10.1016/0024-3795(90)90390-X
- [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] R.S. Varga, Matrix Iterative Analysis, Springer Series in Computational Mathematics (Springer, New York, 2000). Zbl0998.65505

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.