Advances in Numerical Analysis

Volume 2013, Article ID 980615, 6 pages

http://dx.doi.org/10.1155/2013/980615

Research Article

## A New Upper Bound for of a Strictly -Diagonally Dominant -Matrix

^{1}Department of Mathematics and Statistics, Qinghai University for Nationalities, Xining 810007, China^{2}School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received 22 September 2013; Accepted 10 December 2013

Academic Editor: Ting-Zhu Huang

Copyright © 2013 Zhanshan Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- J. M. Varah, “A lower bound for the smallest singular value of a matrix,”
*Linear Algebra and Its Applications*, vol. 11, pp. 3–5, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. S. Varga, “On diagonal dominance arguments for bounding ${\parallel {A}^{-1}\parallel}_{\infty}$,”
*Linear Algebra and Its Applications*, vol. 14, no. 3, pp. 211–217, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G.-H. Cheng and T.-Z. Huang, “An upper bound for ${\parallel {A}^{-1}\parallel}_{\infty}$ of strictly diagonally dominant $M$-matrices,”
*Linear Algebra and Its Applications*, vol. 426, no. 2-3, pp. 667–673, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - P. Wang, “An upper bound for ${\parallel {A}^{-1}\parallel}_{\infty}$ of strictly diagonally dominant $M$-matrices,”
*Linear Algebra and Its Applications*, vol. 431, no. 5–7, pp. 511–517, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - R. A. Horn and C. R. Johnson,
*Topics in Matrix Analysis*, Cambridge University Press, Cambridge, Mass, USA, 1991. View at Publisher · View at Google Scholar · View at MathSciNet - C. R. Johnson, “A Hadamard product involving $M$-matrices,”
*Linear and Multilinear Algebra*, vol. 4, no. 4, pp. 261–264, 1977. View at Google Scholar · View at MathSciNet - M. Fiedler, C. R. Johnson, and T. L. Markham, “A trace inequality for
*M*-matrices and the symmertrizability of a real matrix by a positive diagonal matrix,”*Linear Algebra and Its Applications*, vol. 102, pp. 1–8, 1988. View at Google Scholar - P. N. Shivakumar, J. J. Williams, Q. Ye, and C. A. Marinov, “On two-sided bounds related to weakly diagonally dominant $M$-matrices with application to digital circuit dynamics,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 17, no. 2, pp. 298–312, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - A. Berman and R. J. Plemmons,
*Nonnegative Matrices in the Mathematical Sciences*, Academic Press, New York, NY, USA, 1994. - Y. L. Zhang, H. M. Mo, and J. Z. Liu, “$\alpha $-diagonal dominance and criteria for generalized strictly diagonally dominant matrices,”
*Numerical Mathematics*, vol. 31, no. 2, pp. 119–128, 2009. View at Google Scholar · View at MathSciNet - P. N. Shivakumar and K. H. Chew, “A sufficient condition for nonvanishing of determinants,”
*Proceedings of the American Mathematical Society*, vol. 43, pp. 63–66, 1974. View at Publisher · View at Google Scholar · View at MathSciNet - S. Xu,
*Theory and Methods about Matrix Computation*, Tshua University Press, Beijing, China, 1986.