×

Note on \(Z \)-eigenvalue inclusion theorems for tensors. (English) Zbl 1474.15028

Summary: G. Wang et al. gave four \(Z \)-eigenvalue inclusion intervals for tensors in [Discrete Contin. Dyn. Syst., Ser. B 22, No. 1, 187–198 (2017; Zbl 1362.15014)]. However, these intervals always include zero, and hence could not be used to identify the positive definiteness of a homogeneous polynomial form. In this note, we present a new \(Z \)-eigenvalue inclusion interval with parameters for even-order tensors, which not only overcomes the above shortcomings under certain conditions, but also provides a checkable sufficient condition for the positive definiteness of homogeneous polynomial forms, as well as the asymptotically stability of time-invariant polynomial systems.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A42 Inequalities involving eigenvalues and eigenvectors
15A69 Multilinear algebra, tensor calculus

Citations:

Zbl 1362.15014
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] K. C. Chang; K. J. Pearson; T. Zhang, Some variational principles for \(Z\)-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438, 4166-4182 (2013) · Zbl 1305.15027 · doi:10.1016/j.laa.2013.02.013
[2] C. Deng; H. Li; C. Bu, Brauer-type eigenvalue inclusion sets of stochastic/irreducible tensors and positive definiteness of tensors, Linear Algebra Appl., 556, 55-69 (2018) · Zbl 1395.15008 · doi:10.1016/j.laa.2018.06.032
[3] P. V. D. Driessche, Reproduction numbers of infectious disease models., Infectious Disease Model., 2, 288-303 (2017) · doi:10.1016/j.idm.2017.06.002
[4] O. Duchenne, F. Bach and I. S. Kweon, et al, A tensor-based algorithm for high-order graph matching, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 2383-2395.
[5] J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20, 1290-1301 (2016) · Zbl 1339.15012
[6] J. He; T. Huang, Upper bound for the largest \(Z\)-eigenvalue of positive tensors, Appl. Math. Lett., 38, 110-114 (2014) · Zbl 1314.15016 · doi:10.1016/j.aml.2014.07.012
[7] J. He, Y. Liu and H. Ke, et al, Bounds for the \(Z\)-spectral radius of nonnegative tensors, SpringerPlus, 5 (2016).
[8] J. He; Y. Liu; J. Tian; Z. Zhang, New sufficient condition for the positive definiteness of fourth order tensors, Mathematics, 303, 1-10 (2018) · Zbl 1425.15008 · doi:10.3390/math6120303
[9] E. Kofidis; P. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23, 863-884 (2002) · Zbl 1001.65035 · doi:10.1137/S0895479801387413
[10] T. Kolda; J. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32, 1095-1124 (2011) · Zbl 1247.65048 · doi:10.1137/100801482
[11] C. Li; Y. Li; X. Kong, New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21, 39-50 (2014) · Zbl 1324.15026 · doi:10.1002/nla.1858
[12] C. Li; F. Wang; J. Zhao; Y. Zhu; Y. Li, 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
[13] G. Li; L. Qi; G. Yu, The \(Z\)-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory, Numer. Linear Algebra Appl., 20, 1001-1029 (2013) · Zbl 1313.65081 · doi:10.1002/nla.1877
[14] W. Li; D. Liu; S. W. Vong, \(Z\)-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483, 182-199 (2015) · Zbl 1321.15036 · doi:10.1016/j.laa.2015.05.033
[15] M. Ng; L. Qi; G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31, 1090-1099 (2009) · Zbl 1197.65036 · doi:10.1137/09074838X
[16] Q. Ni; L. Qi; F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. Automat. Control, 53, 1096-1107 (2008) · Zbl 1367.93565 · doi:10.1109/TAC.2008.923679
[17] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40, 1302-1324 (2005) · Zbl 1125.15014 · doi:10.1016/j.jsc.2005.05.007
[18] L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41, 1309-1327 (2006) · Zbl 1121.14050 · doi:10.1016/j.jsc.2006.02.011
[19] L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, 2017. · Zbl 1370.15001
[20] L. Qi; F. Wang; Y. Wang, \(Z\)-eigenvalue methods for a global polynomial optimization problem., Math. Program., 118, 301-316 (2009) · Zbl 1169.90022 · doi:10.1007/s10107-007-0193-6
[21] C. Sang, A new Brauer-type \(Z\)-eigenvalue inclusion set for tensors, Numer. Algorithms, 80, 781-794 (2019) · Zbl 1407.15014 · doi:10.1007/s11075-018-0506-2
[22] Y. Song; L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34, 1581-1595 (2013) · Zbl 1355.15009 · doi:10.1137/130909135
[23] G. Wang; G. Zhou; L. Caccetta, \(Z\)-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst. Ser. B, 22, 187-198 (2017) · Zbl 1362.15014 · doi:10.3934/dcdsb.2017009
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.