On the uniqueness of the positive Z-eigenvector for nonnegative tensors

Abstract The uniqueness of positive Z-eigenvector of nonnegative tensors has many important applications in both theory and practical application. In this paper, we give a sufficient condition that a nonnegative tensor has unique positive Z-eigenvector by using its entries. For the third order transition probabilities tensor, we show that its positive Z-eigenvector corresponding to eigenvalue 1 is unique.

[1]  Michael K. Ng,et al.  Finding the Largest Eigenvalue of a Nonnegative Tensor , 2009, SIAM J. Matrix Anal. Appl..

[2]  Liqun Qi,et al.  Algebraic connectivity of an even uniform hypergraph , 2012, J. Comb. Optim..

[3]  Lek-Heng Lim,et al.  Singular values and eigenvalues of tensors: a variational approach , 2005, 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005..

[4]  Marcello Pelillo,et al.  A generalization of the Motzkin–Straus theorem to hypergraphs , 2009, Optim. Lett..

[5]  Michael K. Ng,et al.  An eigenvalue problem for even order tensors with its applications , 2016 .

[6]  Edward R. Dougherty,et al.  Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks , 2002, Bioinform..

[7]  M. Ng,et al.  On the limiting probability distribution of a transition probability tensor , 2014 .

[8]  Guoyin Li,et al.  Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming , 2013, J. Optim. Theory Appl..

[9]  Kung-Ching Chang,et al.  Some variational principles for Z-eigenvalues of nonnegative tensors , 2013 .

[10]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[11]  R. B. Kellogg,et al.  Uniqueness in the Schauder fixed point theorem , 1976 .

[12]  A. Raftery.  A model for high-order Markov chains , 1985 .

[13]  Kung-Ching Chang,et al.  On the uniqueness and non-uniqueness of the positive Z-eigenvector for transition probability tensors , 2013 .

[14]  Liqun Qi,et al.  Properties of Some Classes of Structured Tensors , 2014, J. Optim. Theory Appl..

[15]  Kung-Ching Chang,et al.  Perron-Frobenius theorem for nonnegative tensors , 2008 .

[16]  Mansoor Saburov,et al.  Ergodicity of nonlinear Markov operators on the finite dimensional space , 2016 .