×

Brauer-type bounds for Hadamard product of nonnegative tensors. (English) Zbl 1445.15018

Summary: In this paper, we establish some Brauer-type bounds for the spectral radius of Hadamard product of two nonnegative tensors based on Brauer-type inclusion set, which are shown to be sharper than the existing bounds established in the literature. The validity of the obtained results is theoretically and numerically tested.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A45 Miscellaneous inequalities involving matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A69 Multilinear algebra, tensor calculus
15A42 Inequalities involving eigenvalues and eigenvectors
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bloy, L.; Verma, R.; Metaxas, D.; Axel, L.; Fichtinger, G.; Székely, G., On computing the underlying fiber directions from the diffusion orientation distribution function, Medical Image Computing and Computer-Assisted Intervention—MICCAI 2008, Part I. Lecture Notes in Comput Sci, Vol 5241, 1-8 (2008), Berlin: Springer, Berlin
[2] Bu, C.; Jin, X.; Li, H.; Deng, C., Brauer-type eigenvalue inclusion sets and the spectral radius of tensors, Linear Algebra Appl, 512, 234-248 (2017) · Zbl 1353.15017 · doi:10.1016/j.laa.2016.09.041
[3] Che, M.; Wei, Y., Theory and Computation of Complex Tensors and its Applications (2020), Singapore: Springer, Singapore · Zbl 07170301
[4] Chen, H.; Qi, L.; Song, Y., Column sufficient tensors and tensor complementarity problems, Front Math China, 13, 255-276 (2018) · Zbl 1418.90253 · doi:10.1007/s11464-018-0681-4
[5] Ding, W.; Wei, Y., Solving multi-linear systems with M-tensors, J Sci Comput, 68, 689-715 (2016) · Zbl 1371.65032 · doi:10.1007/s10915-015-0156-7
[6] Fang, F., Bounds on eigenvalues of Hadamard product and the Fan product of matrices, Linear Algebra Appl, 425, 7-15 (2007) · Zbl 1128.15011 · doi:10.1016/j.laa.2007.03.024
[7] Friedland, S.; Gaubert, S.; Han, L., Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl, 438, 738-749 (2013) · Zbl 1261.15039 · doi:10.1016/j.laa.2011.02.042
[8] Gao, L.; Cao, Z.; Wang, G., Input-to-state stability and integral input-to-state stability for impulsive switched systems with time-delay under asynchronous switching, Nonlinear Anal Hybrid Syst, 34, 248-263 (2019) · Zbl 1434.93104 · doi:10.1016/j.nahs.2019.06.001
[9] Gao, L.; Luo, F.; Yan, Z., Finite-time annular domain stability of impulsive switched systems: mode-dependent parameter approach, Internat J Control, 92, 1381-1392 (2019) · Zbl 1416.93182 · doi:10.1080/00207179.2017.1396360
[10] Horn, R.; Johnson, C., Topics in Matrix Analysis (1985), Cambridge: Cambridge Univ Press, Cambridge · Zbl 0576.15001
[11] Hu, S.; Huang, Z.; Ling, C.; Qi, L., On determinants and eigenvalue theory of tensors, J Symbolic Comput, 50, 508-531 (2013) · Zbl 1259.15038 · doi:10.1016/j.jsc.2012.10.001
[12] Hu, S.; Huang, Z.; Qi, L., Strictly nonnegative tensors and nonnegative tensor partition, Sci China Math, 57, 181-195 (2014) · Zbl 1312.15035 · doi:10.1007/s11425-013-4752-4
[13] Huang, R., Some inequalities for the Hadamard product and the Fan product of matrices, Linear Algebra Appl, 428, 1551-1559 (2008) · Zbl 1163.15017 · doi:10.1016/j.laa.2007.10.001
[14] Jutten, C.; Herault, J., Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture, Signal Processing, 24, 1, 1-10 (1991) · Zbl 0729.73650 · doi:10.1016/0165-1684(91)90079-X
[15] Kolda, T.; Bader, B., Tensor decompositions and applications, SIAM Review, 51, 455-500 (2009) · Zbl 1173.65029 · doi:10.1137/07070111X
[16] Li, C.; Li, Y.; Kong, X., New eigenvalue inclusion sets for tensors, Numer Linear Algebra Appl, 21, 39-50 (2014) · Zbl 1324.15026 · doi:10.1002/nla.1858
[17] Li, Y.; Chen, F.; Wang, D., New lower bounds on eigenvalue of the Hadamard product of an M-matrix and its inverse, Linear Algebra Appl, 430, 1423-1431 (2009) · Zbl 1163.15019 · doi:10.1016/j.laa.2008.11.002
[18] Lim, L. H., Singular values and eigenvalues of tensors: a variational approach, Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Puerto Vallarta, 2005, 129-132 (2005)
[19] Ni, Q.; Qi, L.; Wang, F., An eigenvalue method for testing the positive definiteness of a multivariate form, IEEE Trans Automat Control, 53, 1096-1107 (2008) · Zbl 1367.93565 · doi:10.1109/TAC.2008.923679
[20] Qi, L., Eigenvalues of a real supersymmetric tensor, J Symbolic Comput, 40, 1302-1324 (2005) · Zbl 1125.15014 · doi:10.1016/j.jsc.2005.05.007
[21] Qi, L., Hankel tensors: associated Hankel matrices and Vandermonde decomposition, Commun Math Sci, 13, 113-125 (2015) · Zbl 1331.15020 · doi:10.4310/CMS.2015.v13.n1.a6
[22] Qi, L.; Luo, Z., Tensor Analysis: Spectral Theory and Special Tensors (2017), Philadelphia: SIAM, Philadelphia · Zbl 1370.15001
[23] Sun, L.; Zheng, B.; Zhou, J.; Yan, H., Some inequalities for the Hadamard product of tensors, Linear Multilinear Algebra, 66, 1199-1214 (2018) · Zbl 1391.65090 · doi:10.1080/03081087.2017.1346060
[24] Wang, G.; Wang, Y.; Liu, L., Bound estimations on the eigenvalues for Fan product of M-tensors, Taiwanese J Math, 23, 751-766 (2019) · Zbl 1414.15030 · doi:10.11650/tjm/180905
[25] Wang, G.; Wang, Y.; Zhang, Y., Some inequalities for the Fan product of M-tensors, J Inequal Appl, 2018, 257 (2018) · Zbl 1498.15033 · doi:10.1186/s13660-018-1853-1
[26] Wang, G.; Wang, Y.; Zhang, Y., Brauer-type upper bounds for Z-Spectral radius of weakly symmetric nonnegative tensors, J Math Inequal, 13, 4, 1105-1116 (2019) · Zbl 1432.15009 · doi:10.7153/jmi-2019-13-78
[27] Wang, G.; Zhou, G.; Caccetta, L., Z-eigenvalue inclusion theorems for tensors, Discrete Contin Dyn Syst Ser B, 22, 187-198 (2017) · Zbl 1362.15014
[28] Wang, G.; Zhou, G.; Caccetta, L., Sharp Brauer-type eigenvalue inclusion theorems for tensors, Pac J Optim, 14, 227-244 (2018) · Zbl 1474.15052
[29] Wang, X.; Chen, H.; Wang, Y., Solution structures of tensor complementarity problem, Front Math China, 13, 935-945 (2018) · Zbl 1404.15021 · doi:10.1007/s11464-018-0675-2
[30] Wang, Y.; Zhang, K.; Sun, H., Criteria for strong H-tensors, Front Math China, 11, 577-592 (2016) · Zbl 1381.15019 · doi:10.1007/s11464-016-0525-z
[31] Wang, Y.; Zhou, G.; Caccetta, L., Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer Linear Algebra Appl, 22, 1059-1076 (2015) · Zbl 1374.65105 · doi:10.1002/nla.1996
[32] Wang, Y.; Zhou, G.; Caccetta, L., Nonsingular H-tensor and its criteria, J Ind Manag Optim, 12, 1173-1186 (2016) · Zbl 1364.15019 · doi:10.3934/jimo.2016.12.1173
[33] Yang, Q.; Yang, Y., Further results for Perron-Frobenius theorem for nonnegative tensors II, SIAM J Matrix Anal Appl, 32, 1236-1250 (2011) · Zbl 1426.15011 · doi:10.1137/100813671
[34] Zhou, D.; Chen, G.; Wu, G.; Zhang, X., On some new bounds for eigenvalues of the Hadamard product and the Fan product of matrices, Linear Algebra Appl, 438, 1415-1426 (2013) · Zbl 1262.15022 · doi:10.1016/j.laa.2012.09.013
[35] Zhou, G.; Qi, L.; Wu, S., Efficient algorithms for computing the largest eigenvalue of a nonnegative tensor, Front Math China, 8, 155-168 (2013) · Zbl 1311.65038 · doi:10.1007/s11464-012-0268-4
[36] Zhou, G.; Wang, G.; Qi, L.; Alqahtani, M., A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer Linear Algebra Appl, 25, 2, e2134 (2018) · Zbl 1499.65132 · doi:10.1002/nla.2134
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.