MM-tensors and nonsingular MM-tensors

The M-matrix is an important concept in matrix theory, and has many applications. Recently, this concept has been extended to higher order tensors [18]. In this paper, we establish some important properties of MM-tensors and nonsingular MM-tensors. An MM-tensor is a ZZ-tensor. We show that a ZZ-tensor is a nonsingular MM-tensor if and only if it is semi-positive. Thus, a nonsingular MM-tensor has all positive diagonal entries; an MM-tensor, regarding as the limit of a sequence of nonsingular MM-tensors, has all nonnegative diagonal entries. We introduce even-order monotone tensors and present their spectral properties. In matrix theory, a Z-matrix is a nonsingular M  -matrix if and only if it is monotone. This is no longer true in the case of higher order tensors. We show that an even-order monotone ZZ-tensor is an even-order nonsingular MM-tensor, but not vice versa. An example of an even-order nontrivial monotone ZZ-tensor is also given.