全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用:99美元

查看量下载量

相关文章

更多...

Non-Singularity Conditions for Two Z-Matrix Types

DOI: 10.4236/alamt.2014.42009, PP. 109-119

Keywords: Z-Matrix, M-Matrix, Non-Negative Matrix, Diagonal Dominance

Full-Text   Cite this paper   Add to My Lib

Abstract:

A real square matrix whose non-diagonal elements are non-positive is called a Z-matrix. This paper shows a necessary and sufficient condition for non-singularity of two types of Z-matrices. The first is for the Z-matrix whose row sums are all non-negative. The non-singularity condition for this matrix is that at least one positive row sum exists in any principal submatrix of the matrix. The second is for the Z-matrix \"\"which satisfies \"\"where \"\" . Let \"\" be the ith row and the jth column element of \"\", and \"\" be the jth element of \"\". Let \"\"be a subset of \"\"which is not empty, and \"\" be the complement of \"\" if \"\" is a proper subset. The non-singularity condition for this matrix is \"\"such that \"\"or \"\"such that \"\" for \"\" . Robert Beauwens and Michael Neumann previously presented conditions similar to these conditions. In this paper, we present a different proof and show that these conditions can be also derived from theirs.

References

[1]  Berman, A. and Plemmons, R.J. (1979) Nonnegative Matrices in the Mathematical Sciences. Academic Press, Cambridge.
[2]  Ostrowski, A. (1937-38) über die Determinanten mit überwiegender Hauptdiagonale. Commentarii Mathematici Helvetici, 10, 69-96. http://dx.doi.org/10.1007/BF01214284
[3]  Varga, R.S. (2000) Matrix Iterative Analysis. 2nd Revised and Expanded Edition, Springer, Berlin.
[4]  Nikaido, H. (1968) Convex Structures and Economic Theory. Academic Press, Cambridge.
[5]  DeFranza, J. and Gabliardi, D. (2009) Introduction to Linear Algebra with Applications. International Edition, The McGrow-Hill Higher Education.
[6]  Anton, H. and Rorres, C. (2011) Elementary Linear Algebra with Supplement Applications. International Student Version,10th Edition, John Wiley & Sons, Boston.
[7]  Bretscher, O. (2009) Linear Algebra with Applications. 4th Edition, Pearson Prentice Hall, Upper Saddle River.
[8]  Plemmons, R.J. (1976) M-Matrices Leading to Semiconvergent Splittings. Linear Algebra and its Applications, 15, 243-252. http://dx.doi.org/10.1016/0024-3795(76)90030-6
[9]  Beauwens, R. (1976) Semistrict Diagonal Dominance. SIAM Journal on Numerical Analysis, 13, 109-112.
http://dx.doi.org/10.1137/0713013
[10]  Plemmons, R.J. (1977) M-Matrix Characterizations. 1—Nonsingular M-Matrices. Linear Algebra and Its Applications, 18, 175-188. http://dx.doi.org/10.1016/0024-3795(77)90073-8
[11]  Neumann, M. (1979) A Note on Generalizations of Strict Diagonal Dominance for Real Matrices. Linear Algebra and Its Applications, 26, 3-14. http://dx.doi.org/10.1016/0024-3795(79)90168-X

Full-Text

Contact Us

service@oalib.com

QQ:3279437679

WhatsApp +8615387084133