Some results on the Hadamard product of tensors. (English) Zbl 1418.15019
Summary: In this paper, some fundamental properties of the Hadamard product of tensors are discussed. The closure of several classes of structured tensors under the Hadamard product is studied. As its important application, we present an upper bound on the spectral radius of the Hadamard product of nonnegative tensors. Moreover, we also provide some important inequalities on the spectral radius of the Hadamard products of the Hadamard powers for nonnegative tensors. Some numerical examples are reported to verify the effectiveness of our theoretical results.
MSC:
| 15A69 | Multilinear algebra, tensor calculus |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
References:
| [1] | Qi, L.Q., Luo, Z.Y.: Tensor Analysis: Spectral Theory and Special Tensors. SIAM, Philadelphia (2017) · Zbl 1370.15001 · doi:10.1137/1.9781611974751 |
| [2] | Zhang, L.P., Qi, L.Q., Zhou, G.L.: \[ \cal{M}\] M-tensors and some applications. SIAM J. Matrix Anal. Appl. 35, 437-452 (2014) · Zbl 1307.15034 · doi:10.1137/130915339 |
| [3] | Qi, L.Q.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302-1324 (2005) · Zbl 1125.15014 · doi:10.1016/j.jsc.2005.05.007 |
| [4] | Ding, W.Y., Qi, L.Q., Wei, Y.M.: \[{\cal{M}}\] M-tensors and nonsingular \[{\cal{M}}\] M-tensors. Linear Algebra Appl. 439, 3264-3278 (2013) · Zbl 1283.15074 · doi:10.1016/j.laa.2013.08.038 |
| [5] | Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: CAMSAP’05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-sensor Adaptive Processing, pp. 129-132 (2005) |
| [6] | Li, C.Q., Li, Y.T., Kong, X.: New eigenvalue inclusion sets for tensors. Numer. Linear Algebra Appl. 21, 39-50 (2014) · Zbl 1324.15026 · doi:10.1002/nla.1858 |
| [7] | Chang, K.C., Pearson, K., Zhang, T.: Perron-Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 6, 507-520 (2008) · Zbl 1147.15006 · doi:10.4310/CMS.2008.v6.n2.a12 |
| [8] | Yang, Y.N., Yang, Q.Z.: Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 31, 2517-2530 (2010) · Zbl 1227.15014 · doi:10.1137/090778766 |
| [9] | Li, C.Q., Chen, Z., Li, Y.T.: A new eigenvalue inclusion set for tensors and its applications. Linear Algebra Appl. 481, 36-53 (2015) · Zbl 1320.15020 · doi:10.1016/j.laa.2015.04.023 |
| [10] | Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1985) · Zbl 0729.15001 · doi:10.1017/CBO9780511810817 |
| [11] | Song, Y.Z.: On an inequality for the Hadamard product of an M-matrix and its inverse. Linear Algebra Appl. 305, 99-105 (2000) · Zbl 0946.15014 · doi:10.1016/S0024-3795(99)00224-4 |
| [12] | Liu, Q.B., Chen, G.L.: On two inequalities for the Hadamard product and the Fan product of matrices. Linear Algebra Appl. 431, 974-984 (2009) · Zbl 1183.15017 · doi:10.1016/j.laa.2009.03.049 |
| [13] | Fang, M.Z.: Bounds on the eigenvalues of the 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 |
| [14] | Li, H.B., Huang, T.Z., Shen, Q.S., Hong, L.: Lower bounds for the minimum eigenvalue of Hadamard product of an M-matrix and its inverse. Linear Algebra Appl. 420, 235-247 (2007) · Zbl 1172.15008 · doi:10.1016/j.laa.2006.07.008 |
| [15] | 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 |
| [16] | Zhao, J.X., Sang, C.L., Wang, F.: New inequalities for the Hadamard product of an M-matrix and an inverse M-matrix. J. Inequal. Appl. 368, 1-9 (2015) · Zbl 1335.15028 |
| [17] | Qi, L.Q.: 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 |
| [18] | Li, C.Q., Wang, F., Zhao, J.X., Li, Y.T.: Criterions for the positive definiteness of real supersymmetric tensors. J. Comput. Appl. Math. 255, 1-14 (2014) · Zbl 1291.15065 · doi:10.1016/j.cam.2013.04.022 |
| [19] | Qi, L.Q., Xu, C.Q., Xu, Y.: Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm. SIAM J. Matrix Anal. Appl. 35, 1227-1241 (2014) · Zbl 1317.65114 · doi:10.1137/13092232X |
| [20] | Kannan, M.R., Shaked-Monderer, N., Berman, A.: Some properties of strong \[{\cal{H}}H\]-tensors and general \[{\cal{H}}H\]-tensors. Linear Algebra Appl. 476, 42-55 (2015) · Zbl 1316.15029 · doi:10.1016/j.laa.2015.02.034 |
| [21] | Chen, H.B., Li, G.Y., Qi, L.Q.: Further results on Cauchy tensors and Hankel tensors. Appl. Math. Comput. 275, 50-62 (2016) · Zbl 1410.15048 |
| [22] | Luo, Z.Y., Qi, L.Q.: Completely positive tensors: properties, easily checkable subclasses, and tractable relaxations. SIAM J. Matrix Anal. Appl. 37, 1675-1698 (2016) · Zbl 1349.15026 · doi:10.1137/15M1025220 |
| [23] | Zhou, J., Sun, L.Z., Wei, Y.P., Bu, C.J.: Some characterizations of \[\cal{M}\] M-tensors via digraphs. Linear Algebra Appl. 495, 190-198 (2016) · Zbl 1333.15014 · doi:10.1016/j.laa.2016.01.041 |
| [24] | Sun, L.Z., Zheng, B.D., 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 |
| [25] | Bu, C.J., Zhang, X., Zhou, J., Wang, W.Z., Wei, Y.M.: The inverse, rank and product of tensors. Linear Algebra Appl. 446, 269-280 (2014) · Zbl 1286.15030 · doi:10.1016/j.laa.2013.12.015 |
| [26] | Li, C.Q., Li, Y.T.: Double \[{\cal{B}}B\]-tensors and quasi-double \[{\cal{B}}B\]-tensors. Linear Algebra Appl. 466, 343-356 (2015) · Zbl 1303.15034 · doi:10.1016/j.laa.2014.10.027 |
| [27] | Song, Y.S., Qi, L.Q.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854-873 (2015) · Zbl 1390.15085 · doi:10.1007/s10957-014-0616-5 |
| [28] | Qi, L.Q., Song, Y.S.: An even order symmetric \[{\cal{B}}B\]-tensor is positive definite. Linear Algebra Appl. 457, 303-312 (2014) · Zbl 1295.15017 · doi:10.1016/j.laa.2014.05.026 |
| [29] | Shao, J.Y.: A general product of tensors with applications. Linear Algebra Appl. 439, 2350-2366 (2013) · Zbl 1283.15076 · doi:10.1016/j.laa.2013.07.010 |
| [30] | Hu, S.L., Huang, Z.H., Qi, L.Q.: Strictly nonnegative tensors and nonnegative tensor partition. Sci. China Math. 57, 181-195 (2014) · Zbl 1312.15035 · doi:10.1007/s11425-013-4752-4 |
| [31] | Zhao, L.L., Liu, Q.B.: Some inequalities on the spectral radius of matrices. J. Inequal. Appl. 5, 1-12 (2018) · Zbl 1386.15013 |
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.
