MTensors and Some Applications
Abstract
We introduce Mtensors. This concept extends the concept of Mmatrices. We denote Ztensors as the tensors with nonpositive offdiagonal entries. We show that Mtensors must be Ztensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric Mtensor must be nonnegative. A symmetric Mtensor 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 Heigenvalue. We show that a Ztensor is an Mtensor if and only if all its H+ eigenvalues are nonnegative. Some further spectral properties of Mtensors are given. We also introduce strong Mtensors, and some corresponding conclusions are given. In particular, we show that all Heigenvalues of strong Mtensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with Ztensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form.
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Copyright © 2014 Society for Industrial and Applied Mathematics
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