<bold>?-tensors and nonsingular </bold><bold>?-tensors</bold>

doi:10.1007/s11464-015-0495-6

Front. Math. China

2016, Vol. 11

Issue (3) : 557-575 https://doi.org/10.1007/s11464-015-0495-6

RESEARCH ARTICLE

?-tensors and nonsingular ?-tensors

Xuezhong WANG^1,²,Yimin WEI^1,^*(

)

1. School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2. School of Mathematics and Statistics, Hexi University, Zhangye 734000, China

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Abstract

The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order ?-tensors. In this paper, we establish important properties of diagonally dominant tensors and ?-tensors. Distributions of eigenvalues of nonsingular symmetric ?-tensors are given. An ?₊-tensor is semi-positive, which enlarges the area of semi-positive tensor from ?-tensor to ?₊-tensor. The spectral radius of Jacobi tensor of a nonsingular (resp. singular) ?-tensor is less than (resp. equal to) one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular ?-tensor if and only if all of its principal sub-tensors are nonsingular ?-tensors. An irreducible tensor Ais an ?-tensor if and only if it is quasi-diagonally dominant.

Keywords Diagonally dominant irreducible diagonally dominant ?-tensor nonsingular

Corresponding Author(s): Yimin WEI

Issue Date: 17 May 2016

Cite this article:

Xuezhong WANG,Yimin WEI. ?-tensors and nonsingular ?-tensors[J]. Front. Math. China, 2016, 11(3): 557-575.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-015-0495-6
https://academic.hep.com.cn/fmc/EN/Y2016/V11/I3/557

1	Berman A, Plemmons R J. Nonnegative Matrices in the Mathematical Sciences. Philadelphia: SIAM, 1994 https://doi.org/10.1137/1.9781611971262
2	Chang K C, Pearson K, Zhang T. Perron-Frobenius theorem for nonnegative tensors. Commun Math Sci, 2008, 6: 507–520 https://doi.org/10.4310/CMS.2008.v6.n2.a12
3	Chang K C, Pearson K, Zhang T. On eigenvalue problems of real symmetric tensors. J Math Anal Appl, 2009, 350: 416–422 https://doi.org/10.1016/j.jmaa.2008.09.067
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 https://doi.org/10.1137/100807120
5	Ding W, Qi L, Wei Y. ℳ-tensors and nonsingular ℳ-tensors. Linear Algebra Appl, 2013, 439: 3264–3278 https://doi.org/10.1016/j.laa.2013.08.038
6	Hu S, Huang Z, Qi L. Strictly nonnegative tensors and nonnegative tensor partition. Sci China Math, 2014, 57: 181–195 https://doi.org/10.1007/s11425-013-4752-4
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 https://doi.org/10.1137/S0895479801387413
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 https://doi.org/10.1016/j.cam.2013.04.022
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 https://doi.org/10.1016/j.jsc.2005.05.007
11	Qi L. Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl, 2013, 439: 228–238 https://doi.org/10.1016/j.laa.2013.03.015
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 https://doi.org/10.1016/j.laa.2015.02.034
13	Varga R S. Matrix Iterative Analysis. Berlin: Springer-Verlag, 2000 https://doi.org/10.1007/978-3-642-05156-2
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 https://doi.org/10.1137/100813671
16	Yang Y, Yang Q. Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31: 2517–2530 https://doi.org/10.1137/090778766
17	Yuan P, You L. On the similarity of tensors. Linear Algebra Appl, 2014, 458: 534–541 https://doi.org/10.1016/j.laa.2014.06.026
18	Zhang L, Qi L, Zhou G. ℳ-tensors and some applications. SIAM J Matrix Anal Appl, 2014, 35: 437–452 https://doi.org/10.1137/130915339

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