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doi:10.1007/s11464-016-0519-x

Front. Math. China

2016, Vol. 11

Issue (3) : 661-678 https://doi.org/10.1007/s11464-016-0519-x

RESEARCH ARTICLE

Criterions for identifying H-tensors

Ruijuan ZHAO¹,Lei GAO^2,³,Qilong LIU²,Yaotang LI^2,^*(

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1. School of Information Engineering, Lanzhou University of Finance and Economics, Lanzhou 730020, China
2. School of Mathematics and Statistics, Yunnan University, Kunming 650091, China
3. Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji 721013, China

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Abstract

Some new criteria for identifying H-tensors are obtained. As applications, some sufficient conditions of the positive definiteness for an evenorder real symmetric tensor are given, as well as a new eigenvalue inclusion region for tensors is established. It is proved that the new eigenvalue inclusion region is tighter than that of Y. Yang and Q. Yang [SIAM J.Matrix Anal. Appl., 2010, 31: 2517–2530]. Numerical examples are reported to demonstrate the corresponding results.

Keywords H-Tensor')" href="#">

-Tensor real symmetric tensor positive definite eigenvalue inclusion set

Corresponding Author(s): Yaotang LI

Issue Date: 17 May 2016

Cite this article:

Ruijuan ZHAO,Lei GAO,Qilong LIU, et al. Criterions for identifying H-tensors[J]. Front. Math. China, 2016, 11(3): 661-678.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0519-x
https://academic.hep.com.cn/fmc/EN/Y2016/V11/I3/661

1	Anderson B, Bose N, Jury E. Output feedback stabilization and related problemssolutions via decision methods. IEEE Trans Automat Control AC, 1975, 20: 55–66
2	Bose N, Kamat P. Algorithm for stability test of multidimensional filters. IEEE Trans Acoust Speech Signal Process ASSP, 1974, 22: 307–314 https://doi.org/10.1109/TASSP.1974.1162592
3	Bose N, Modarressi A. General procedure for multivariable ploynomial positivity with control applications. IEEE Trans Autom control AC, 1976, 21: 696–701
4	Bose N, Newcomb R. Tellegons theorem and multivariable realizability theory. Int J Electron, 1974, 36: 417–425 https://doi.org/10.1080/00207217408900421
5	Cartwright D, Sturmfels B. The number of eigenvalues of a tensor. Linear Algebra Appl, 2013, 438: 942–952 https://doi.org/10.1016/j.laa.2011.05.040
6	Chang K C, Pearson K, Zhang T. Perron Frobenius theorem for nonnegative tensors. Commun Math Sci, 2008, 6: 507–520 https://doi.org/10.4310/CMS.2008.v6.n2.a12
7	Ding W, Qi L, Wei Y. M-tensors and nonsingular M-tensors. Linear Algebra Appl, 2013, 439: 3264–3278 https://doi.org/10.1016/j.laa.2013.08.038
8	Fu M. Comments on A procedure for the positive definiteness of forms of even-order. IEEE Trans Automat Control, 1988, 43: 1430
9	Hasan M, Hasan A. A procedure for the positive definiteness of forms of even-order. IEEE Trans Automat Control AC, 1996, 41: 615–617 https://doi.org/10.1109/9.489287
10	Hu S, Huang Z, Ling C, Qi L. E-determinants of tensors. J Symb Comput, 2013, 50:508–531 https://doi.org/10.1016/j.jsc.2012.10.001
11	Kofidis E, Regalia PA. On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J Matrix Anal Appl, 2002, 23: 863–884 https://doi.org/10.1137/S0895479801387413
12	Kolda T, Mayo J. Shifted power method for computing tensor eigenpairs. SIAM J Matrix Anal Appl, 2011, 32: 1095–1124 https://doi.org/10.1137/100801482
13	Ku W. Explicit criterion for the positive definiteness of a general quartic form. IEEE Trans Automat control AC, 1965, 10: 372–373
14	Lathauwer L, Moor B, Vandewalle J. A multilinear singular value decomposition. SIAM J Matrix Anal Appl, 2000, 21: 1253–1278 https://doi.org/10.1137/S0895479896305696
15	Li C, Li Y, Kong X. New eigenvalue inclusion sets for tensor. Numer Linear Algebra Appl, 2014, 21: 39–50 https://doi.org/10.1002/nla.1858
16	Li C, Wang F, Zhao J, Zhu Y, Li Y. Criterions for the positive definiteness of real supersymmetric tensors. J Comput Appl Math, 2014, 255: 1–14 https://doi.org/10.1016/j.cam.2013.04.022
17	Lim L. Singular values and eigenvalues of tensors: A variational approach. In: IEEE CAMSAP 2005: First International Workshop on Computational Advances in Multi-Sensor Adaptive Processing. New York: IEEE, 2005, 129–132
18	Ni G, Qi L, Wang F, Wang Y. The degree of the E-characteristic polynomial of an even order tensor. J Math Anal Appl, 2007, 329: 1218–1229 https://doi.org/10.1016/j.jmaa.2006.07.064
19	Ni Q, Qi L, Wang F. An eigenvalue method for the positive definiteness identification problem. IEEE Trans Automat Control, 2008, 53: 1096–1107 https://doi.org/10.1109/TAC.2008.923679
20	Qi L. Eigenvalues of a real supersymetric tensor. J Symbolic Comput, 2005, 40: 1302–1324 https://doi.org/10.1016/j.jsc.2005.05.007
21	Qi L. Eigenvalues and invariants of tensors. J Math Anal Appl, 2007, 325: 1363–1377 https://doi.org/10.1016/j.jmaa.2006.02.071
22	Qi L, Wang F, Wang Y. Z-eigenvalue methods for a global polynomial optimization problem. Math Program, 2009, 118: 301–316 https://doi.org/10.1007/s10107-007-0193-6
23	Qi L, Wang Y, Wu E. D-eigenvalues of diffusion kurtosis tensors. J Comput Appl Math, 2008, 221: 150–157 https://doi.org/10.1016/j.cam.2007.10.012
24	Wang F, Qi L. Comments on ‘Explicit criterion for the positive definiteness of a general quartic form’. IEEE Trans Automat Control, 2005, 50: 416–418 https://doi.org/10.1109/TAC.2005.843851
25	Wang Y, Qi L, Zhang X. A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor. Numerical Linear Algebra Appl, 2009, 16: 589–601 https://doi.org/10.1002/nla.633
26	Yang Y, Yang Q. Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31: 2517–2530 https://doi.org/10.1137/090778766
27	Zhang L, Qi L, Zhou G. M-tensors and some applications. SIAM J Matrix Anal Appl, 2014, 32: 437–452 https://doi.org/10.1137/130915339

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