Further results on B-tensors with application to location of real eigenvalues

Abstract

In this paper, we give a further study on B-tensors and introduce doubly B-tensors that contain B-tensors. We show that they have similar properties, including their decompositions and strong relationship with strictly (doubly) diagonally dominated tensors. As an application, the properties of B-tensors are used to localize real eigenvalues of some tensors, which would be very useful in verifying the positive semi-definiteness of a tensor.

The location of H-eigenvalues of real even order symmetry tensors

“…Using the desired decomposition, they proved that any even order symmetric B-tensor is positive definite. Ye et al [20] had a further study on the location of real eigenvalues of some class of tensors. They defined another two classes of tensors called B-tensors and doubly B-tensors.…”

“…In this paper, we will have a further study on the location of Downloaded by [University of California Santa Barbara] at 22:52 24 June 2016 H-eigenvalues of real even order symmetry tensors based on these foundation works. [18][19][20] The outline of this paper is as follows. In Section 2, we provide some preliminaries on tensors.…”

“…[20] Let A = (a i 1 ...i m ) ∈ T (R n , m) be a tensor with the kth row tensor A k , k ∈ [n] and A = (a i 1 ...i m ) ∈ T (R n , m) be a tensor with the kth row tensor sign(a k...k )A k , k ∈ [n].Then, A is called a B-tensor if and only if A is a B-tensor. Downloaded by [University of California Santa Barbara] at 22:52 24 June 2016…”