Eigenvalue bounds of third-order tensors via the minimax eigenvalue of symmetric matrices

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.

Quasi-Double Diagonally Dominant H-Tensors and the Estimation Inequalities for the Spectral Radius of Nonnegative Tensors

“…> 0, the M-tensor is also an important application [9]. The estimation of the upper and lower bounds for the spectral radius of a nonnegative tensor is an important element in the study of the spectral problem of nonnegative tensors [14,15], and the application of the relation between the M-tensor and the nonnegative tensor gives an estimate of the upper and lower bounds for the spectral radius of the nonnegative tensor. By analyzing the tensor structure, two classes of quasi-double diagonally dominant tensors are given in this paper, and they are proved to be H-tensors; at the same time, an inequality is given for the estimation of the upper and lower bounds for the spectral radius of the nonnegative tensor.…”

Some improvements on the Ky Fan theorem for tensors

“…Qi (2005) gave an eigenvalue inclusion set for real symmetric tensors, which is a generalization of the well-known Gershgorin set of matrices (Horn and Johnson 2012). Subsequently, due to its fundamental applications in various fields, many researchers are interested in investigating eigenvalue inclusion regions for tensors, e.g., Bu et al (2017), Li and Li (2016a), Li et al (2014Li et al ( , 2020, Xu et al (2019), etc. In 1958, Fan (1958 obtained the famous Ky Fan theorem that plays an important role in the study of the nonnegative eigenvalue problem.…”

Properties and calculation for <i>C</i>-eigenvalues of a piezoelectric-type tensor

“…The existence and properties on C-eigenpairs of a piezoelectric-type tensor is discussed in [4]. Recently, Che et al [3], Li et al [16], Li et al [17], Liu et al [21], Wang et al [28], Xiong and Liu [30] considered the C-eigenvalue localization and presented many C-eigenvalue inclusion intervals to locate all C-eigenvalues of a piezoelectric-type tensor. However, there is not an efficient solution method to obtain all C-eigenvalues of a piezoelectric-type tensor.…”

“…Now, we locate all C-eigenvalues of the above eight piezoelectric tensors by using the intervals in Theorems 2.1, 2.2, 2.3, 2.4 and 2.5, Theorems 2.2 and 2.4 in [3], Theorem 2 of [16], Theorem 7 of [17], Theorem 2.1 of [21], Theorem 2.1 in [28] and Theorem 2.1 of [30]. Numerical results are showed in Table 1.…”