Z-eigenvalues based structured tensors: \(\mathcal {M}_z\)-tensors and strong \(\mathcal {M}_z\)-tensors

  • Changxin Mo
  • Chaoqian Li
  • Xuezhong Wang
  • Yimin WeiEmail author


The positive (semi-)definiteness of even-order tensors has been widely studied in these years due to its applications in various aspects, such as spectral hypergraph theory, automatic control, polynomial theory, stochastic process, magnetic resonance imaging and so on. It has been shown that \(\mathcal {M}\)-tensors, \(\mathcal {B}\)-tensors, \(\mathcal {H}\)-tensors, Hilbert tensors and stochastic tensors can be positive definite under proper conditions. However, there are still many positive definite tensors that can not be determined by the above criteria. In this paper, we provide a new class of positive definite tensors whose non-diagonal entries can be positive compared to (strong) \(\mathcal {M}\)-tensors, and we call it strong \(\mathcal {M}_z\)-tensors, which can arise from discretizing differential equations, since it is based on Z-eigenvalues rather than H-eigenvalues traditionally. Moreover, we show that an even-order (strong) \(\mathcal {M}\)-tensor must be an (a strong) \(\mathcal {M}_z\)-tensor, which reflects the inclusion relationship between even-order \(\mathcal {M}\)-tensors and \(\mathcal {M}_z\)-tensors. We also introduce (strong) \(\mathcal {H}_z\)-tensors, as a generalization of (strong) \(\mathcal {M}_z\)-tensors, and its positive semi-definiteness (positive definiteness) has been studied. Finally, some conditions are given for a tensor to be an (a strong) \(\mathcal {M}_z\)-tensor and we use it to study the stability of a high-order nonlinear system.


\(\mathcal {M}_z\)-tensor Z-eigenvalue \(\mathcal {H}_z\)-tensor \(\mathcal {M}\)-tensor Positive definiteness Stability of nonlinear system 

Mathematics Subject Classification

15A18 15A69 



The authors are indebted to the editor and two referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of this paper. Also, the first author would like to thank Dr. Maolin Che, Dr. Weiyang Ding and Dr. Liping Zhang for their useful discussions and valuable suggestions.


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© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsYunnan UniversityKunmingPeople’s Republic of China
  3. 3.School of Mathematics and StatisticsHexi UniversityZhangyePeople’s Republic of China
  4. 4.School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied MathematicsFudan UniversityShanghaiPeople’s Republic of China

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