Criterions for identifying H-tensors

Ruijuan ZHAO, Lei GAO, Qilong LIU, Yaotang LI

PDF(170 KB)
PDF(170 KB)
Front. Math. China ›› 2016, Vol. 11 ›› Issue (3) : 661-678. DOI: 10.1007/s11464-016-0519-x
RESEARCH ARTICLE
RESEARCH ARTICLE

Criterions for identifying H-tensors

Author information +
History +

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 / real symmetric tensor / positive definite / eigenvalue inclusion set

Cite this article

Download citation ▾
Ruijuan ZHAO, Lei GAO, Qilong LIU, Yaotang LI. Criterions for identifying H-tensors. Front. Math. China, 2016, 11(3): 661‒678 https://doi.org/10.1007/s11464-016-0519-x

References

[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
CrossRef Google scholar
[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
CrossRef Google scholar
[5]
Cartwright D, Sturmfels B. The number of eigenvalues of a tensor. Linear Algebra Appl, 2013, 438: 942–952
CrossRef Google scholar
[6]
Chang K C, Pearson K, Zhang T. Perron Frobenius theorem for nonnegative tensors. Commun Math Sci, 2008, 6: 507–520
CrossRef Google scholar
[7]
Ding W, Qi L, Wei Y. M-tensors and nonsingular M-tensors. Linear Algebra Appl, 2013, 439: 3264–3278
CrossRef Google scholar
[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
CrossRef Google scholar
[10]
Hu S, Huang Z, Ling C, Qi L. E-determinants of tensors. J Symb Comput, 2013, 50:508–531
CrossRef Google scholar
[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
CrossRef Google scholar
[12]
Kolda T, Mayo J. Shifted power method for computing tensor eigenpairs. SIAM J Matrix Anal Appl, 2011, 32: 1095–1124
CrossRef Google scholar
[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
CrossRef Google scholar
[15]
Li C, Li Y, Kong X. New eigenvalue inclusion sets for tensor. Numer Linear Algebra Appl, 2014, 21: 39–50
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[19]
Ni Q, Qi L, Wang F. An eigenvalue method for the positive definiteness identification problem. IEEE Trans Automat Control, 2008, 53: 1096–1107
CrossRef Google scholar
[20]
Qi L. Eigenvalues of a real supersymetric tensor. J Symbolic Comput, 2005, 40: 1302–1324
CrossRef Google scholar
[21]
Qi L. Eigenvalues and invariants of tensors. J Math Anal Appl, 2007, 325: 1363–1377
CrossRef Google scholar
[22]
Qi L, Wang F, Wang Y. Z-eigenvalue methods for a global polynomial optimization problem. Math Program, 2009, 118: 301–316
CrossRef Google scholar
[23]
Qi L, Wang Y, Wu E. D-eigenvalues of diffusion kurtosis tensors. J Comput Appl Math, 2008, 221: 150–157
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[26]
Yang Y, Yang Q. Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31: 2517–2530
CrossRef Google scholar
[27]
Zhang L, Qi L, Zhou G. M-tensors and some applications. SIAM J Matrix Anal Appl, 2014, 32: 437–452
CrossRef Google scholar

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
AI Summary AI Mindmap
PDF(170 KB)

Accesses

Citations

Detail

Sections
Recommended

/