<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="script">M</mml:mi></mml:math>-tensors and nonsingular<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="script">M</mml:mi></mml:math>-tensors

Ding, Weiyang; Qi, Liqun; Wei, Yimin

doi:10.1016/j.laa.2013.08.038

Cited by 252 publications

(149 citation statements)

References 20 publications

(31 reference statements)

Supporting

Mentioning

144

Contrasting

Order By: Relevance

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php conditions for a Z-tensor A to be a strong M -tensor? We know that [11,Theorem 3] gives a positive answer for the first condition. The second question is still open.…”

Section: Discussionmentioning

confidence: 99%

$M$-Tensors and Some Applications

Zhang

Qi²,

Zhou³

2014

SIAM J. Matrix Anal. & Appl.

Self Cite

246

155

View full text Add to dashboard Cite

Abstract. We introduce M -tensors. This concept extends the concept of M -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. 1. Introduction. Tensors are increasingly ubiquitous in various areas of applied, computational, and industrial mathematics and have wide applications in data analysis and mining, information science, signal/image processing, and computational biology as well [5,9,15,17]. A tensor can be regarded as a higher-order generalization of a matrix, which takes the form Key words. M -tensors, H

show abstract

Section: Discussionmentioning

confidence: 99%

$M$-Tensors and Some Applications

Zhang

Qi²,

Zhou³

2014

SIAM J. Matrix Anal. & Appl.

Self Cite

246

155

View full text Add to dashboard Cite

show abstract

“…Definition 2 [5,18] We call a tensor A as an M -tensor, if there exists a nonnegative tensor B and a positive real number η ρ(B) such that…”

Section: Lemma 1 [10]mentioning

confidence: 99%

“…Zhang et al [18] extended M -matrices to M -tensors and studied their properties. Ding et al [5] gave ten equivalent conditions for M -tensors and extended the other two definitions of nonsingular M -matrices, semi-positivity and monotonicity [1,13], to higher-order tensors.…”

mentioning

confidence: 99%

ℋ-tensors and nonsingular ℋ-tensors

Wang

Wei

2015

Front. Math. China

Self Cite

View full text Add to dashboard Cite

The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order H -tensors. In this paper, we establish important properties of diagonally dominant tensors and H -tensors. Distributions of eigenvalues of nonsingular symmetric H -tensors are given. An H + -tensor is semi-positive, which enlarges the area of semi-positive tensor from M -tensor to H + -tensor. The spectral radius of Jacobi tensor of a nonsingular (resp. singular) H -tensor is less than (resp. equal to) one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular H -tensor if and only if all of its principal sub-tensors are nonsingular H -tensors. An irreducible tensor A is an H -tensor if and only if it is quasi-diagonally dominant.

show abstract

“…Recently, by introducing the definition of H-tensor [11,12], Li et al [12] provided a practical sufficient condition for identifying the positive definiteness of an even-order symmetric tensor (see Proposition 1.2).…”

Section: Proposition 11 ([1]mentioning

confidence: 99%