?-tensors and nonsingular ?-tensors

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 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 Ais an ℋ-tensor if and only if it is quasi-diagonally dominant.

Keywords

Diagonally dominant / irreducible diagonally dominant / ℋ-tensor / nonsingular

References

[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

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg