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dc.contributor.authorZhang, L.
dc.contributor.authorQi, L.
dc.contributor.authorZhou, Guanglu
dc.identifier.citationZhang, 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.

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.

dc.publisherSociety for Industrial and Applied Mathematics
dc.titleM-Tensors and Some Applications
dc.typeJournal Article
dcterms.source.titleSIAM Journal on Matrix Analysis and Applications

Copyright © 2014 Society for Industrial and Applied Mathematics

curtin.departmentDepartment of Mathematics and Statistics
curtin.accessStatusOpen access

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