-
Previous Article
Semidefinite programming via image space analysis
- JIMO Home
- This Issue
- Next Article
Nonsingular $H$-tensor and its criteria
1. | School of Management Science, Qufu Normal University, Rizhao Shandong, 276800, China |
2. | Department of Mathematics and Statistics, Curtin University of Technology, West Australia, WA 6102 |
3. | Department of Mathematics and Statistics, Curtin University, Perth, Western Australia, 6102, Australia |
References:
[1] |
J. Brachat, P. Comon, B. Mourrain and E. Tsigaridas, Symmetric tensor decomposition,, Linear Algebra Appl., 433 (2010), 1851.
doi: 10.1016/j.laa.2010.06.046. |
[2] |
K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors,, Commun. Math. Science, 6 (2008), 507.
doi: 10.4310/CMS.2008.v6.n2.a12. |
[3] |
H. B. Chen and L. Q. Qi, Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors,, J. Industrial and Management Optim., 11 (2015), 1263.
doi: 10.3934/jimo.2015.11.1263. |
[4] |
A. Cichocki, R. Zdunek, A. H. Phan and S. Amari, Nonnegative Matrix and Tensor Factorizations,, John Wiley & Sons, (2009).
doi: 10.1002/9780470747278. |
[5] |
L. Cvetkovic, V. Kostic and R. S. Varga, A new Geršgorin-type eigenvalue inclusion set,, Elec. Trans. Numer. Anal., 18 (2004), 73.
|
[6] |
L. Cvetkovic and V. Kostic, New criteria for identifying $H$-matrices,, J Comput. Appl. Math., 180 (2005), 265.
doi: 10.1016/j.cam.2004.10.017. |
[7] |
L. De Lathauwer, B. De Moor and J. Vandewalle, A multilinear singular value decomposition,, SIAM J. Matrix Anal. Appl., 21 (2000), 1253.
doi: 10.1137/S0895479896305696. |
[8] |
W. Y. Ding, L. Q. Qi and Y. M. Wei, $M$-tensors and nonsingular $M$-tensors,, Linear Algebra Appl., 439 (2013), 3264.
doi: 10.1016/j.laa.2013.08.038. |
[9] |
S. Gandy, B. Recht and I. Yamada, Tensor completion and low-$n$-rank tensor recovery via convex optimization,, Inverse Problems, 27 (2011).
doi: 10.1088/0266-5611/27/2/025010. |
[10] |
R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, (1985).
doi: 10.1017/CBO9780511810817. |
[11] |
S. L. Hu and L. Qi, Algebraic connectivity of an even uniform hypergraph,, J. Combination Optim., 24 (2012), 564.
doi: 10.1007/s10878-011-9407-1. |
[12] |
S. L. Hu, Z. H. Huang, C. Ling and L. Q. Qi, On determinants and eigenvalue theory of tensors,, J. Symbolic Comput., 50 (2013), 508.
doi: 10.1016/j.jsc.2012.10.001. |
[13] |
S. L. Hu and L. Q. Qi, Algebraic connectivity of an even uniform hypergraph,, J. Combinatorial Optim., 24 (2012), 564.
doi: 10.1007/s10878-011-9407-1. |
[14] |
M. R. Kannan, N. Shaked-Monderer and A. Berman, Some properties of strong H-tensors and general H-tensors,, Linear Algebra Appl., 476 (2015), 42.
doi: 10.1016/j.laa.2015.02.034. |
[15] |
E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors,, SIAM J. Matrix Anal. Appl., 23 (2002), 863.
doi: 10.1137/S0895479801387413. |
[16] |
T. G. Kolda and B. W. Bader, Tensor decompositions and applications,, SIAM Review, 51 (2009), 455.
doi: 10.1137/07070111X. |
[17] |
T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenvalues,, SIAM J. Matrix Anal. Appl, 32 (2011), 1095.
doi: 10.1137/100801482. |
[18] |
C. Q. Li, F. Wang, J. X. Zhao, Y. Zhu and Y. T. Li, Criterions for the positive definiteness of real supersymmetric tensors,, J. Comput. Appl. Math., 255 (2014), 1.
doi: 10.1016/j.cam.2013.04.022. |
[19] |
Y. Y. Liu and F. H. Shang, An efficient matrix factorization method for tensor completion,, IEEE Signal Processing Letters, 20 (2013), 307.
doi: 10.1109/LSP.2013.2245416. |
[20] |
L. H. Lim, Singular value and and eigenvalue of tensors, a variational approach,, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in multiSensor Adaptive Processing, (2005), 129. Google Scholar |
[21] |
J. Liu, P. Musialski, P. Wonka and J. P. Ye, Tensor completion for estimating missing values in visual data,, IEEE Trans. on Pattern Anal. Machine Intelligence, 35 (2013), 208. Google Scholar |
[22] |
M. Moakher, On the averaging of symmetric positive-definite tensors,, J. Elasticity, 82 (2006), 273.
doi: 10.1007/s10659-005-9035-z. |
[23] |
M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor,, SIAM J. Matrix Anal. Appl., 31 (2009), 1090.
doi: 10.1137/09074838X. |
[24] |
Q. Ni, L. Qi and F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form,, IEEE Trans. on Auto. Control, 53 (2008), 1096.
doi: 10.1109/TAC.2008.923679. |
[25] |
C. L. Nikias and J. M. Mendel, Signal processing with higher-order spectra,, IEEE Signal Processing Magazine, 10 (1993), 10.
doi: 10.1109/79.221324. |
[26] |
L. Oeding and G. Ottaviani, Eigenvectors of tensors and algorithms for Waring decomposition,, J. Symbolic Comput., 54 (2013), 9.
doi: 10.1016/j.jsc.2012.11.005. |
[27] |
A. M. Ostrowski, Über die Determinaanten mit überwiegender Hauptdiagonale,, Comment Math. Helv., 10 (1937), 69.
doi: 10.1007/BF01214284. |
[28] |
L. Q. Qi, Eigenvalues of a real supersymmetric tensor,, J. Symbolic Comput., 40 (2005), 1302.
doi: 10.1016/j.jsc.2005.05.007. |
[29] |
L. Qi, J. Y. Shao and Q. Wang, Regular uniform hypergraphs, $s$-cycles, $s$-paths and their largest Laplacian H-eigenvalues,, Linear Algebra Appl., 443 (2014), 215.
doi: 10.1016/j.laa.2013.11.008. |
[30] |
L. Qi, C. Xu and Y. Xu, Nonnegative tensor factorization, completely positive tensors and an Hierarchically elimination algorithm,, SIAM J. Matrix Anal. Appl., 35 (2014), 1227.
doi: 10.1137/13092232X. |
[31] |
Y. S. Song and L. Qi, Necessary and sufficient conditions for copositive tensors,, Linear and Multilinear Algebra, 63 (2015), 120.
doi: 10.1080/03081087.2013.851198. |
[32] |
Y. Song and L. Q. Qi, Infinite and finite dimensional Hilbert tensors,, Linear Algebra Appl., 451 (2014), 1.
doi: 10.1016/j.laa.2014.03.023. |
[33] |
Y. S. Song and L. Q. Qi, Properties of some classes of structured tensors,, J. Optim. Theory Appl., 165 (2015), 854.
doi: 10.1007/s10957-014-0616-5. |
[34] |
Y. Yang and Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors,, SIAM J. Matrix Anal. Appl., 31 (2010), 2517.
doi: 10.1137/090778766. |
[35] |
P. Yuan and L. You, Some remarks on $P,P_0, B$ and $B_0$ tensors,, Linear Algebra Appl., 459 (2014), 511.
doi: 10.1016/j.laa.2014.07.043. |
[36] |
L. P. Zhang, L. Q. Qi and G. L. Zhou, $M$-tensors and some applications,, SIAM J. Matrix Anal. Appl., 35 (2014), 437.
doi: 10.1137/130915339. |
[37] |
X. Z. Zhang, C. Ling and L. Qi, The best rank-1 approximation of a symmetric tensor and related spherical optimization problems,, SIAM J. Matrix Anal. Appl., 33 (2012), 806.
doi: 10.1137/110835335. |
show all references
References:
[1] |
J. Brachat, P. Comon, B. Mourrain and E. Tsigaridas, Symmetric tensor decomposition,, Linear Algebra Appl., 433 (2010), 1851.
doi: 10.1016/j.laa.2010.06.046. |
[2] |
K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors,, Commun. Math. Science, 6 (2008), 507.
doi: 10.4310/CMS.2008.v6.n2.a12. |
[3] |
H. B. Chen and L. Q. Qi, Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors,, J. Industrial and Management Optim., 11 (2015), 1263.
doi: 10.3934/jimo.2015.11.1263. |
[4] |
A. Cichocki, R. Zdunek, A. H. Phan and S. Amari, Nonnegative Matrix and Tensor Factorizations,, John Wiley & Sons, (2009).
doi: 10.1002/9780470747278. |
[5] |
L. Cvetkovic, V. Kostic and R. S. Varga, A new Geršgorin-type eigenvalue inclusion set,, Elec. Trans. Numer. Anal., 18 (2004), 73.
|
[6] |
L. Cvetkovic and V. Kostic, New criteria for identifying $H$-matrices,, J Comput. Appl. Math., 180 (2005), 265.
doi: 10.1016/j.cam.2004.10.017. |
[7] |
L. De Lathauwer, B. De Moor and J. Vandewalle, A multilinear singular value decomposition,, SIAM J. Matrix Anal. Appl., 21 (2000), 1253.
doi: 10.1137/S0895479896305696. |
[8] |
W. Y. Ding, L. Q. Qi and Y. M. Wei, $M$-tensors and nonsingular $M$-tensors,, Linear Algebra Appl., 439 (2013), 3264.
doi: 10.1016/j.laa.2013.08.038. |
[9] |
S. Gandy, B. Recht and I. Yamada, Tensor completion and low-$n$-rank tensor recovery via convex optimization,, Inverse Problems, 27 (2011).
doi: 10.1088/0266-5611/27/2/025010. |
[10] |
R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, (1985).
doi: 10.1017/CBO9780511810817. |
[11] |
S. L. Hu and L. Qi, Algebraic connectivity of an even uniform hypergraph,, J. Combination Optim., 24 (2012), 564.
doi: 10.1007/s10878-011-9407-1. |
[12] |
S. L. Hu, Z. H. Huang, C. Ling and L. Q. Qi, On determinants and eigenvalue theory of tensors,, J. Symbolic Comput., 50 (2013), 508.
doi: 10.1016/j.jsc.2012.10.001. |
[13] |
S. L. Hu and L. Q. Qi, Algebraic connectivity of an even uniform hypergraph,, J. Combinatorial Optim., 24 (2012), 564.
doi: 10.1007/s10878-011-9407-1. |
[14] |
M. R. Kannan, N. Shaked-Monderer and A. Berman, Some properties of strong H-tensors and general H-tensors,, Linear Algebra Appl., 476 (2015), 42.
doi: 10.1016/j.laa.2015.02.034. |
[15] |
E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors,, SIAM J. Matrix Anal. Appl., 23 (2002), 863.
doi: 10.1137/S0895479801387413. |
[16] |
T. G. Kolda and B. W. Bader, Tensor decompositions and applications,, SIAM Review, 51 (2009), 455.
doi: 10.1137/07070111X. |
[17] |
T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenvalues,, SIAM J. Matrix Anal. Appl, 32 (2011), 1095.
doi: 10.1137/100801482. |
[18] |
C. Q. Li, F. Wang, J. X. Zhao, Y. Zhu and Y. T. Li, Criterions for the positive definiteness of real supersymmetric tensors,, J. Comput. Appl. Math., 255 (2014), 1.
doi: 10.1016/j.cam.2013.04.022. |
[19] |
Y. Y. Liu and F. H. Shang, An efficient matrix factorization method for tensor completion,, IEEE Signal Processing Letters, 20 (2013), 307.
doi: 10.1109/LSP.2013.2245416. |
[20] |
L. H. Lim, Singular value and and eigenvalue of tensors, a variational approach,, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in multiSensor Adaptive Processing, (2005), 129. Google Scholar |
[21] |
J. Liu, P. Musialski, P. Wonka and J. P. Ye, Tensor completion for estimating missing values in visual data,, IEEE Trans. on Pattern Anal. Machine Intelligence, 35 (2013), 208. Google Scholar |
[22] |
M. Moakher, On the averaging of symmetric positive-definite tensors,, J. Elasticity, 82 (2006), 273.
doi: 10.1007/s10659-005-9035-z. |
[23] |
M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor,, SIAM J. Matrix Anal. Appl., 31 (2009), 1090.
doi: 10.1137/09074838X. |
[24] |
Q. Ni, L. Qi and F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form,, IEEE Trans. on Auto. Control, 53 (2008), 1096.
doi: 10.1109/TAC.2008.923679. |
[25] |
C. L. Nikias and J. M. Mendel, Signal processing with higher-order spectra,, IEEE Signal Processing Magazine, 10 (1993), 10.
doi: 10.1109/79.221324. |
[26] |
L. Oeding and G. Ottaviani, Eigenvectors of tensors and algorithms for Waring decomposition,, J. Symbolic Comput., 54 (2013), 9.
doi: 10.1016/j.jsc.2012.11.005. |
[27] |
A. M. Ostrowski, Über die Determinaanten mit überwiegender Hauptdiagonale,, Comment Math. Helv., 10 (1937), 69.
doi: 10.1007/BF01214284. |
[28] |
L. Q. Qi, Eigenvalues of a real supersymmetric tensor,, J. Symbolic Comput., 40 (2005), 1302.
doi: 10.1016/j.jsc.2005.05.007. |
[29] |
L. Qi, J. Y. Shao and Q. Wang, Regular uniform hypergraphs, $s$-cycles, $s$-paths and their largest Laplacian H-eigenvalues,, Linear Algebra Appl., 443 (2014), 215.
doi: 10.1016/j.laa.2013.11.008. |
[30] |
L. Qi, C. Xu and Y. Xu, Nonnegative tensor factorization, completely positive tensors and an Hierarchically elimination algorithm,, SIAM J. Matrix Anal. Appl., 35 (2014), 1227.
doi: 10.1137/13092232X. |
[31] |
Y. S. Song and L. Qi, Necessary and sufficient conditions for copositive tensors,, Linear and Multilinear Algebra, 63 (2015), 120.
doi: 10.1080/03081087.2013.851198. |
[32] |
Y. Song and L. Q. Qi, Infinite and finite dimensional Hilbert tensors,, Linear Algebra Appl., 451 (2014), 1.
doi: 10.1016/j.laa.2014.03.023. |
[33] |
Y. S. Song and L. Q. Qi, Properties of some classes of structured tensors,, J. Optim. Theory Appl., 165 (2015), 854.
doi: 10.1007/s10957-014-0616-5. |
[34] |
Y. Yang and Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors,, SIAM J. Matrix Anal. Appl., 31 (2010), 2517.
doi: 10.1137/090778766. |
[35] |
P. Yuan and L. You, Some remarks on $P,P_0, B$ and $B_0$ tensors,, Linear Algebra Appl., 459 (2014), 511.
doi: 10.1016/j.laa.2014.07.043. |
[36] |
L. P. Zhang, L. Q. Qi and G. L. Zhou, $M$-tensors and some applications,, SIAM J. Matrix Anal. Appl., 35 (2014), 437.
doi: 10.1137/130915339. |
[37] |
X. Z. Zhang, C. Ling and L. Qi, The best rank-1 approximation of a symmetric tensor and related spherical optimization problems,, SIAM J. Matrix Anal. Appl., 33 (2012), 806.
doi: 10.1137/110835335. |
[1] |
Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617 |
[2] |
Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]