M-tensors and nonsingular M-tensors

By Weiyang Ding, Liqun Qi, Yimin Wei. 2013

321 citations recorded by the source corpus.

Abstract 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; an M -tensor, regarding as the limit of a sequence 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.

Subfields: Computational Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/8ae62a3d5f684e79ad1cbdbc05dedaf3ad952eaa

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In Computational Mathematics

1. Chaoqian Li
2. Yaotang Li
3. Yisheng Song
4. Liqun Qi
5. Haibin Chen
6. Guanglu Zhou
7. Aiquan Jiao
8. L. Caccetta

In Computational Theory and Mathematics

1. Chaoqian Li
2. Yaotang Li
3. Yisheng Song
4. Guanglu Zhou
5. Liqun Qi
6. L. Caccetta
7. Haibin Chen
8. Yiju Wang

Citations in the corpus

• Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors (2014, background): “In recent years, a lot of research papers on structured tensors are presented [2, 3, 4, 8, 13, 14, 15, 17, 18, 21, 22], which studied M-tensors, circulant tensors, completely positive tensors, Hankel tensors, Hilbert tensors, P tensors and B tensors.”
• An even order symmetric B tensor is positive definite (2014, background): “The properties of M and strong M tensors may be found in [4,6,29].”
• Some remarks on P, P0, B and B0 tensors (2014, background): “It was proved in [9] that a diagonally dominated Z tensor is an M tensor, and a strictly diagonally dominated Z tensor is a strongM tensor ([1, 9]), where strongM tensors are called nonsingular tensors ([1]), and Laplacian tensors of uniform hypergraphs which is defined as a natural extension of Lap”
• Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors (2014, background): “For example, Zhang, Qi and Zhou [16] and Ding, Qi and Wei [17] for Mtensors, Song and Qi [18] for P-(P0)tensors and B-(B0)tensors, Qi and Song [19] for positive (semi-)definition of B-(B0)tensors, Song and Qi [20] for infinite and finite dimensional Hilbert tensors, Song and Qi [22] for structure pr”
• An S-type eigenvalue localization set for tensors (2015, methodology): “Introduction Eigenvalue problems of tensors have become an important topic of study in numerical multilinear algebra, and they have a wide range of practical applications; see [2, 3, 4, 5, 7, 8, 9, 10, 11, 17, 18, 19, 21, 22, 24, 25].”
• The Sparse Solution to $\mathcal{KS}$-Tensor Complementarity Problems (2022, unknown): “2 [37] Let A ∈ R be a Z-tensor, then the following propositions are equivalent: [(i)] 1.”
• Z-tensors and complementarity problems (2015, background): “As A has the S -tensor property (see Theorem 4.3), it follows from [7, Theorem 3] and [7, Proposition 5] that there exists a positive diagonal matrix Now we claim that for any given q ∈ R n , TCP( ¯ A , q ) has a unique solution.”
• A new eigenvalue inclusion set for tensors and its applications (2015, methodology): “Like nonnegative matrices, we can by nonnegative tensors define strong M -tensors appeared in [29], which are called nonsingular M -tensors in [30].”
• Double B-tensors and quasi-double B-tensors (2014, background): “B-tensors are linked with positive definite tensors and M-tensors, which are useful in automatical control, magnetic resonance imaging and spectral hypergraph theory [2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 23, 24, 25, 26, 28, 30, 31, 33].”
• Nonsingular $H$-tensor and its criteria (2016, methodology): “To explore criteria with weaker conditions for nonsingular H-tensor, we give the following definition [8].”