Frontiers of Mathematics in China >
ℋ-tensors and nonsingular ℋ-tensors
Received date: 21 Jan 2015
Accepted date: 31 Jul 2015
Published date: 17 May 2016
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The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order ℋ-tensors. In this paper, we establish important properties of diagonally dominant tensors and ℋ-tensors. Distributions of eigenvalues of nonsingular symmetric ℋ-tensors are given. An ℋ+-tensor is semi-positive, which enlarges the area of semi-positive tensor from ℳ-tensor to ℋ+-tensor. The spectral radius of Jacobi tensor of a nonsingular (resp. singular) ℋ-tensor is less than (resp. equal to) one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular ℋ-tensor if and only if all of its principal sub-tensors are nonsingular ℋ-tensors. An irreducible tensor is an ℋ-tensor if and only if it is quasi-diagonally dominant.
Xuezhong WANG , Yimin WEI . ℋ-tensors and nonsingular ℋ-tensors[J]. Frontiers of Mathematics in China, 2016 , 11(3) : 557 -575 . DOI: 10.1007/s11464-015-0495-6
| 1 |
Berman A, Plemmons R J. Nonnegative Matrices in the Mathematical Sciences. Philadelphia: SIAM, 1994
|
| 2 |
Chang K C, Pearson K, Zhang T. Perron-Frobenius theorem for nonnegative tensors. Commun Math Sci, 2008, 6: 507–520
|
| 3 |
Chang K C, Pearson K, Zhang T. On eigenvalue problems of real symmetric tensors. J Math Anal Appl, 2009, 350: 416–422
|
| 4 |
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
|
| 5 |
Ding W, Qi L, Wei Y. ℳ-tensors and nonsingular ℳ-tensors. Linear Algebra Appl, 2013, 439: 3264–3278
|
| 6 |
Hu S, Huang Z, Qi L. Strictly nonnegative tensors and nonnegative tensor partition. Sci China Math, 2014, 57: 181–195
|
| 7 |
Kofidis E, Regalia P A. On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J Matrix Anal Appl, 2001/02, 23: 863–884
|
| 8 |
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
|
| 9 |
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
|
| 10 |
Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324
|
| 11 |
Qi L. Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl, 2013, 439: 228–238
|
| 12 |
Rajesh-Kanan M, Shaked-Monderer N, Berman A. Some properties of strong ℋ-tensors and general ℋ-tensors. Linear Algebra Appl, 2015, 476: 42–55
|
| 13 |
Varga R S. Matrix Iterative Analysis. Berlin: Springer-Verlag, 2000
|
| 14 |
Wang Y, Zhou G, Caccetta L. Nonsingular ℋ-tensors and their criteria. J Ind Manag Optim (to appear)
|
| 15 |
Yang Q, Yang Y. Further results for Perron-Frobenius theorem for nonnegative tensors II. SIAM J Matrix Anal Appl, 2011, 32: 1236–1250
|
| 16 |
Yang Y, Yang Q. Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31: 2517–2530
|
| 17 |
Yuan P, You L. On the similarity of tensors. Linear Algebra Appl, 2014, 458: 534–541
|
| 18 |
Zhang L, Qi L, Zhou G. ℳ-tensors and some applications. SIAM J Matrix Anal Appl, 2014, 35: 437–452
|
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