Geodesic
Intervals of certain classes of Z-matrices
Discussiones Mathematicae. General Algebra and Applications, Tome 34 (2014) no. 1, pp. 85-93
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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.
Keywords: interval matrix, M-matrix, N-matrix, N₀-matrix, nonnegativity 
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Kannan, M.; Sivakumar, K. Intervals of certain classes of Z-matrices. Discussiones Mathematicae. General Algebra and Applications, Tome 34 (2014) no. 1, pp. 85-93. http://geodesic.mathdoc.fr/item/DMGAA_2014_34_1_a5/

[1] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (SIAM, Philadelphia, 1994).

[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

[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

[9] R.S. Varga, Matrix Iterative Analysis, Springer Series in Computational Mathematics (Springer, New York, 2000).