Li-Ping ZhangDepartment of Mathematical Sciences, Tsinghua UniversityLiqun QiDepartment of Applied Mathematics, Hong Kong Polytechnic UniversityGuanglu ZhouDepartment of Mathematics and Statistics, Curtin University, Perth, Australia
Numerical Linear Algebramathscidoc:1804.26001
SIAM Journal on Matrix Analysis and Applications, 35, (2), 437-452, 2014.4
We introduceM-tensors. This concept extends the concept ofM-matrices. We denote
Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M-tensors must be Ztensors
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 Mtensor
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.