Some Criteria for $${\varvec{\mathcal {H}}}$$ H -Tensors
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Some Criteria for H‑Tensors Guangbin Wang1 · Fuping Tan2 Received: 2 August 2019 / Revised: 10 November 2019 / Accepted: 14 December 2019 © Shanghai University 2020
Abstract The H-tensor is a new developed concept in tensor analysis and it is an extension of the M -tensor. In this paper, we present some criteria for identifying nonsingular H-tensors and give two numerical examples. Keywords H-tensor · Generalized diagonally dominant · M-tensor Mathematics Subject Classification 15A48 · 15A69
1 Introduction A high order tensor is a multi-way array whose entries are addressed via multiple indices in the following form: ) ( 𝐀 = ai1 i2 ⋯im , ai1 i2 ⋯im ∈ ℝ, ij = 1, 2, ⋯ , nj , j = 1, 2, ⋯ , m, where ℝ is the set of real numbers. If n1 = n2 = ⋯ = nm , then 𝐀 is called a square tensor; otherwise, it is called a rectangular tensor. Tensors are higher-order extensions of matrices, and they have wide applications in signal and image processing, continuum physics, higher-order statistics, blind source separation, and especially in exploratory multi-way data analysis [4]. Hence, tensor analysis and computing have received much attention of researchers in the recent decade. The H-tensor which contains the M-tensor as special cases plays an important role in identifying positive definiteness of even-order real symmetric tensors [2, 3, 5, 11]. In [1, 6, 7, 9, 10, 12], the authors presented some criteria for H-tensors. In [8], the authors gave some properties and conditions for H-tensors and nonsingular H-tensors.
* Fuping Tan [email protected] Guangbin Wang [email protected] 1
Department of Mathematics, Qingdao Agricultural University, Qingdao 266061, Shandong Province, China
2
Department of Mathematics, Shanghai University, Shanghai 200444, China
13
Vol.:(0123456789)
Communications on Applied Mathematics and Computation
For an m-th order n-dimensional tensor and a vector X ∈ ℝn , 𝐀X m−1 is a vector in ℝn with entries
( m−1 ) 𝐀X = i
∑
n,n,⋯,n
aii2 i3 ⋯im Xi2 Xi3 ⋯ Xim , i = 1, 2, ⋯ , n,
i2 ,i3 ,⋯,im =1
and 𝐀X m is a scalar with
∑
n,n,⋯,n
𝐀X m =
ai1 i2 ⋯im Xi1 Xi2 Xi3 ⋯ Xim .
i1 ,i2 ,⋯,im =1
We use the Kronecker delta function
𝛿i1 i2 ⋯im =
{
1, if i1 = i2 = ⋯ = im , 0, otherwise,
and use I = (𝛿i1 i2 ⋯im ) to denote the m-th order n-dimensional identity tensor.
2 Preliminaries In this section, we first give some preliminaries developed on tensor analysis and computing, and then discuss some properties of nonsingular H-tensors. For a real m-th order n-dimensional tensor 𝐀 and a scalar 𝜆 ∈ ℂ , if there exists a nonzero vector X ∈ ℂn , such that
𝐀X m−1 = 𝜆X [m−1] , where X [m−1] ∈ ℂn with (X [m−1] )i = Xim−1 , i = 1, 2, ⋯ , n , then 𝜆 is said to be an eigenvalue of the tensor 𝐀 and X an eigenvector associated with the eigenvalue 𝜆 . In particular, if X is real, then 𝜆 is also real, and we say ( 𝜆, X ) is an H-eigenpair of the tensor 𝐀 . The largest modulus of eigenvalue of the tensor 𝐀 is called the spectral radius of the tensor 𝐀 and we denote it by 𝜌(𝐀).
Definition 2.1 [11] A tensor 𝐀 i
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