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    M-Tensors and Some Applications

    200863_133060_SIMAX-M-Tensor-ZQZ-2014.pdf (254.2Kb)
    Access Status
    Open access
    Authors
    Zhang, L.
    Qi, L.
    Zhou, Guanglu
    Date
    2014
    Type
    Journal Article
    
    Metadata
    Show full item record
    Citation
    Zhang, L. and Qi, L. and Zhou, G. 2014. M-Tensors and Some Applications. SIAM Journal on Matrix Analysis and Applications. 35 (2): pp. 437-452.
    Source Title
    SIAM Journal on Matrix Analysis and Applications
    DOI
    10.1137/130915339
    ISSN
    08954798
    School
    Department of Mathematics and Statistics
    Remarks

    Copyright © 2014 Society for Industrial and Applied Mathematics

    URI
    http://hdl.handle.net/20.500.11937/35236
    Collection
    • Curtin Research Publications
    Abstract

    We introduce M-tensors. This concept extends the concept of M-matrices. We denote Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M-tensors must be Z-tensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric M-tensor must be nonnegative. A symmetric M-tensor is copositive. Based on the spectral theory of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an M-tensor is its smallest H+ -eigenvalue and also is its smallest H-eigenvalue. We show that a Z-tensor is an M-tensor if and only if all its H+ -eigenvalues are nonnegative. Some further spectral properties of M-tensors are given. We also introduce strong M-tensors, and some corresponding conclusions are given. In particular, we show that all H-eigenvalues of strong M-tensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with Z-tensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form.

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