Eigenvalue bounds of third-order tensors via the minimax eigenvalue of symmetric matrices
- PDF / 391,133 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 34 Downloads / 239 Views
Eigenvalue bounds of third-order tensors via the minimax eigenvalue of symmetric matrices Shigui Li1 · Zhen Chen1 · Chaoqian Li2 · Jianxing Zhao3 Received: 17 April 2020 / Revised: 23 June 2020 / Accepted: 2 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract Upper and lower bounds for H -eigenvalues, Z -spectral radius and C-spectral radius of a thirdorder tensor are given by the minimax eigenvalue of symmetric matrices extracted from this given tensor. As applications, a sufficient condition for third-order nonsingular M-tensors and some valid sufficient conditions for the uniqueness and solvability of the solutions to multi-linear systems, tensor complementarity problems and non-homogeneous systems are proposed. Keywords H -eigenvalues · Z -eigenvalues · C-eigenvalues · Nonsingular M-tensors · Tensor complementarity problems · Piezoelectric-type tensors Mathematics Subject Classification 15A18 · 15A42 · 15A69
1 Introduction Third-order tensors have received much attention and have extensive applications in physics, engineering and scientific computing (Grozdanov and Kaplis 2016; Sørensen and De Lath-
Communicated by Yimin Wei.
B
Zhen Chen [email protected] Shigui Li [email protected] Chaoqian Li [email protected] Jianxing Zhao [email protected]
1
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, People’s Republic of China
2
School of Mathematics and Statistics, Yunnan University, Yunnan 650091, People’s Republic of China
3
College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, People’s Republic of China 0123456789().: V,-vol
123
217
Page 2 of 14
S. Li et al..
auwer 2015; Royer et al. 2011). Commonly, third-order tensors can be thought of as 3-dimensional arrays of data, which is used to process data in practice (Kilmer et al. 2013; Padhy and Dandapat 2017). Eigenproblems of tensors have become a vibrant applications in applied mathematics branch and numerical multilinear algebra (Qi and Teo 2003; Qi 2007; Ding and Wei 2015). There are several kinds of eigenvalues for tensors, such as M-eigenvalues of the elasticity tensors related to the strong ellipticity conditions in nonlinear mechanics (Qi et al. 2018), D-eigenvalues of diffusion kurtosis tensors in diffusion kurtosis imaging (Qi et al. 2008) and B-eigenvalues of symmetric tensors (Cui et al. 2014). In the paper, we consider bounds for H -eigenvalues, Z -eigenvalues and C-eigenvalues of third-order tensors. The H -eigenvalues and Z -eigenvalues of tensor are introduced independently by Qi (2005) and Lim (2005). Let A be a real mth-order n-dimensional tensor, Rn be the set of all real vectors of dimension n, and [n] := {1, 2, . . . , n}. Definition 1 (Qi 2005; Lim 2005) A real number λ is called an H -eigenvalue of a real mthorder n-dimensional tensor A, if there exists nonzero vector x ∈ Rn satisfying Axm−1 = λx[m−1] ,
where Axm−1 and x[m−1] are two real vectors with their ith components aii2 ...im xi2 . . . xim and (x[
Data Loading...
