Frontiers of Mathematics in China >
A new definition of geometric multiplicity of eigenvalues of tensors and some results based on it
Received date: 04 Apr 2014
Accepted date: 04 Jul 2014
Published date: 24 Jun 2015
Copyright
We give a new definition of geometric multiplicity of eigenvalues of tensors, and based on this, we study the geometric and algebraic multiplicity of irreducible tensors’ eigenvalues. We get the result that the eigenvalues with modulus ρ() have the same geometric multiplicity. We also prove that twodimensional nonnegative tensors’ geometric multiplicity of eigenvalues is equal to algebraic multiplicity of eigenvalues.
Yiyong LI , Qingzhi YANG , Yuning YANG . A new definition of geometric multiplicity of eigenvalues of tensors and some results based on it[J]. Frontiers of Mathematics in China, 2015 , 10(5) : 1123 -1146 . DOI: 10.1007/s11464-015-0412-z
| 1 |
Bulò S R, Pelillo M. A generalization of the Motzkin-Straus theorem to hypergraphs. Optim Lett, 2009, 3: 287-295
|
| 2 |
Bulò S R, Pelillo M. New bounds on the clique number of graphs based on spectral hypergraph theory. In: Stützle T, ed. Learning and Intelligent Optimization. Berlin: Springer-Verlag, 2009, 45-48
|
| 3 |
Chang K C. A nonlinear Krein Rutman theorem. J Syst Sci Complex, 2009, 22: 542-554
|
| 4 |
Chang K C, Pearson K, Zhang T. Perron Frobenius Theorem for non-negative tensors. Commun Math Sci, 2008, 6: 507-520
|
| 5 |
Chang K C, Pearson K, Zhang T. On eigenvalue problems of real symmetric tensors. J Math Anal Appl, 2009, 350: 416-422
|
| 6 |
Chang K C, Pearson K, Zhang T. Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors. SIAM J Matrix Anal Appl, 2011, 32: 806-819
|
| 7 |
Chang K C, Qi L, Zhang T. A survey on the spectral theory of nonnegative tensors. Numer Linear Algebra Appl, 2013, 20: 891-912
|
| 8 |
Chang K C, Zhang T. Multiplicity of singular values for tensors. Commun Math Sci, 2009, 7: 611-625
|
| 9 |
Friedland S, Gaubert S, Han L. Perron-Frobenius theorem for nonnegative multilinear forms and extensions. Linear Algebra Appl, 2013, 438: 738-749
|
| 10 |
Hu S, Huang Z H, Qi L. Strictly nonnegative tensors and nonnegative tensor partition. Sci China Math, 2014, 57: 181-195
|
| 11 |
Lim L H. Singular values and eigenvalues of tensors: a variational approach. Proceedings of the IEEE InternationalWorkshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005, 1: 129-132
|
| 12 |
Ng M, Qi L, Zhou G. Finding the largest eigenvalue of a non-negative tensor. SIAM J Matrix Anal Appl, 2009, 31: 1090-1099
|
| 13 |
Pearson K. Essentially positive tensors. Int J Algebra, 2010, 4: 421-427
|
| 14 |
Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302-1324
|
| 15 |
Qi L, Sun W, Wang Y. Numerical multilinear algebra and its applications. Front Math China, 2007, 2: 501-526
|
| 16 |
Shao J. A general product of tensors with applications. Linear Algebra and Its Applications, 2013, 439: 2350-2366
|
| 17 |
Varga R. Matrix Iterative Analysis. Berlin: Springer-Verlag, 2000
|
| 18 |
Yang Q, Yang Y. Further results for Perron-Frobenius Theorem for nonnegative tensors II. SIAM J Matrix Anal Appl, 2011, 32: 1236-1250
|
| 19 |
Yang Y, Yang Q. Further results for Perron-Frobenius Theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31: 2517-2530
|
| 20 |
Yang Y, Yang Q. Geometric simplicity of the spectral radius of nonnegative irreducible tensors. Front Math China, 2013, 8(1): 129-140
|
/
| 〈 |
|
〉 |