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 M-tensors and nonsingular M-tensors. An M-tensor is a Z-tensor. We show that a Z-tensor is a nonsingular M-tensor if and only if it is semi-positive. Thus, a nonsingular M-tensor has all positive diagonal entries; and an M-tensor, regarding as the limitation of a series of nonsingular M-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 Z-tensor is an even-order nonsingular M-tensor but not vice versa. An example of an even-order nontrivial monotone Z-tensor is also given.

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SummaryThis paper is contributed to a fast algorithm for Hankel tensor–vector products. First, we explain the necessity of fast algorithms for Hankel and block Hankel tensor–vector products by sketching the algorithm for both one‐dimensional and multi‐dimensional exponential data fitting. For proposing the fast algorithm, we define and investigate a special class of Hankel tensors that can be diagonalized by the Fourier matrices, which is called anti‐circulant tensors. Then, we obtain a fast algorithm for Hankel tensor–vector products by embedding a Hankel tensor into a larger anti‐circulant tensor. The computational complexity is about scriptO(m2nlogmn) for a square Hankel tensor of order m and dimension n, and the numerical examples also show the efficiency of this scheme. Moreover, the block version for multi‐level block Hankel tensors is discussed. Copyright © 2015 John Wiley & Sons, Ltd.

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