Some results on Brauer-type and Brualdi-type eigenvalue inclusion sets for tensors
- PDF / 432,096 Bytes
- 13 Pages / 439.37 x 666.142 pts Page_size
- 73 Downloads / 159 Views
(2019) 38:74
Some results on Brauer-type and Brualdi-type eigenvalue inclusion sets for tensors Yangyang Xu1 · Bing Zheng1 · Ruijuan Zhao1 Received: 17 December 2018 / Revised: 28 February 2019 / Accepted: 7 March 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this paper, we establish the comparison between Brauer-type eigenvalue inclusion set given by Bu et al. (Linear Algebra Appl 512:234–248, 2017) and a Brualdi-type eigenvalue inclusion set given by Bu et al. (Linear Algebra Appl 480:168–175, 2015), and investigate the relationship of some Brauer-type eigenvalue inclusion sets provided by Li et al. (Linear Multilinear Algebra 64:587–601, 2016a) and Bu et al. (2017). In particular, we provide a sufficient condition such that the Brualdi-type eigenvalue inclusion set is tighter than some Brauer-type eigenvalue inclusion sets. Moreover, some sufficient conditions depending on entries of the given tensor for identifying strong M-tensor and the positive definiteness of an even-order real symmetric tensor are presented. To verify our theoretical results and show their effectiveness, some numerical examples are given. Keywords Brauer-type · Brualdi-type · The eigenvalue inclusion set · Strong M-tensor · Positive definiteness Mathematics Subject Classification 15A69 · 15A18
1 Introduction Let C(R) be the set of all complex(real) numbers, and [n] = {1, 2, . . . , n}. An order m dimension n complex(real) tensor A = (ai1 i2 ···im ) is a multidimensional array with n m entries ai1 i2 ···im ∈ C(R),
Communicated by Jinyun Yuan.
B
Bing Zheng [email protected] Yangyang Xu [email protected] Ruijuan Zhao [email protected]
1
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China
123
74
Page 2 of 13
Y. Xu et al.
where i j ∈ [n], j ∈ [m]. Especially, when m = 2, A is an n-by-n matrix. A tensor A is called symmetric Qi (2005) if its entries are invariant under any permutation of their indices. Moreover, an order m dimension n tensor I = (δi1 i2 ···im ) is called the identity tensor Qi (2005), where 1, if i 1 = i 2 = · · · = i m , δi1 i2 ···im = 0, otherwise. Qi (2005) and Lim (2005) defined an eigenvalue-eigenvector pair of a tensor, respectively. For an order m dimension n tensor A, a pair (λ, x) ∈ C × (Cn \{0}) is called an eigenvalueeigenvector pair of A if Ax m−1 = λx [m−1] ,
where Ax m−1 is an n-dimensional column vector whose ith entry is aii2 ···im xi2 · · · xim , (Ax m−1 )i = i 2 ,...,i m ∈[n]
and x [m−1] = (x1m−1 , x2m−1 , . . . , xnm−1 )T . In this case, we call λ an H-eigenvalue of A and x an H-eigenvector of A associated with λ if both λ and x are real. Furthermore, the spectral radius of A is defined as ρ(A) = max{|λ| : λ ∈ σ (A)}, where σ (A) is the set of all eigenvalues of A. With the introduction of tensor eigenvalue-eigenvector pair, the spectral theory of higher order tensors has become a hot topic in numerical multi-linear algebra, and has a strong practical background and a wide range of theoretical a
Data Loading...
