Article
Full entry |
PDF
(1.1 MB)
Feedback
PDF
(1.1 MB)
Feedback
Keywords:
lower bound sequence; Hadamard product; $M$-matrix; doubly stochastic matrix; $S$-type eigenvalue inclusion set
lower bound sequence; Hadamard product; $M$-matrix; doubly stochastic matrix; $S$-type eigenvalue inclusion set
Summary:
We propose a lower bound sequence for the minimum eigenvalue of Hadamard product of an $M$-matrix and its inverse, in terms of an $S$-type eigenvalues inclusion set and inequality scaling techniques. In addition, it is proved that the lower bound sequence converges. Several numerical experiments are given to demonstrate that the lower bound sequence is sharper than some existing ones in most cases.
References:
[1] Berman, A., Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics 9. SIAM, Philadelphia (1994). DOI 10.1137/1.9781611971262 | MR 1298430 | Zbl 0815.15016
[2] Chen, F.-B.: New inequalities for the Hadamard product of an $M$-matrix and its inverse. J. Inequal. Appl. 2015 (2015), Article ID 35, 12 pages. DOI 10.1186/s13660-015-0555-1 | MR 3304942 | Zbl 1307.15005
[3] Cheng, G., Tan, Q., Wang, Z.: Some inequalities for the minimum eigenvalue of the Hadamard product of an $M$-matrix and its inverse. J. Inequal. Appl. 2013 (2013), Article ID 65, 9 pages. DOI 10.1186/1029-242X-2013-65 | MR 3031582 | Zbl 1282.15010
[4] Fiedler, M., Johnson, C. R., Markham, T. L., Neumann, M.: A trace inequality for $M$-matrices and the symmetrizability of a real matrix by a positive diagonal matrix. Linear Algebra Appl. 71 (1985), 81-94. DOI 10.1016/0024-3795(85)90237-X | MR 0813035 | Zbl 0597.15016
[5] Fiedler, M., Markham, T. L.: An inequality for the Hadamard product of an $M$-matrix and inverse $M$-matrix. Linear Algebra Appl. 101 (1988), 1-8. DOI 10.1016/0024-3795(88)90139-5 | MR 0941292 | Zbl 0648.15009
[6] Fiedler, M., Pták, V.: Diagonally dominant matrices. Czech. Math. J. 17 (1967), 420-433. DOI 10.21136/CMJ.1967.100787 | MR 0215869 | Zbl 0178.03402
[7] Huang, Z., Wang, L., Xu, Z.: Some new estimations for the Hadamard product of a nonsingular $M$-matrix and its inverse. Math. Inequal. Appl. 20 (2017), 661-682. DOI 10.7153/mia-20-44 | MR 3653913 | Zbl 1377.15010
[8] Li, H.-B., Huang, T.-Z., Shen, S.-Q., Li, H.: Lower bounds for the minimum eigenvalue of Hadamard product of an $M$-matrix and its inverse. Linear Algebra Appl. 420 (2007), 235-247. DOI 10.1016/j.laa.2006.07.008 | MR 2277644 | Zbl 1172.15008
[9] Sinkhorn, R., Knopp, P.: Concerning nonnegative matrices and doubly stochastic matrices. Pac. J. Math. 21 (1967), 343-348. DOI 10.2140/pjm.1967.21.343 | MR 0210731 | Zbl 0152.01403
[10] Varga, R. S.: Geršgorin and His Circles. Springer Series in Computational Mathematics 36. Springer, Berlin (2004). DOI 10.1007/978-3-642-17798-9 | MR 2093409 | Zbl 1057.15023
[11] Zhao, J., Wang, F., Sang, C.: Some inequalities for the minimum eigenvalue of the Hadamard product of an $M$-matrix and an inverse $M$-matrix. J. Inequal. Appl. 2015 (2015), Article ID 92, 9 pages. DOI 10.1186/s13660-015-0611-x | MR 3318918 | Zbl 1308.15003
[12] Zhou, D., Chen, G., Wu, G., Zhang, X.: Some inequalities for the Hadamard product of an $M$-matrix and an inverse $M$-matrix. J. Inequal. Appl. 2013 (2013), Article ID 16, 10 pages. DOI 10.1186/1029-242X-2013-16 | MR 3017347 | Zbl 1281.15017
Search
Partner of
