Chaoqian Li

An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors

A new eigenvalue localization set for tensors is given, and proved to be tighter than those in [Qi L. Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput. 2005;40:1302–1324] and [Li CQ, Li YT, Kong X. New eigenvalue inclusion sets for tensors. Numer. Linear Algebra Appl. 2014;21:39–50]. Based on this set, we give two checkable sufficient conditions of the positive definiteness of tensors, and two checkable sufficient conditions of the positive semi-definiteness of tensors.

A new Brauer-type eigenvalue localization set for tensors

A new Brauer-type eigenvalue localization set for tensors is given. We show that this set is tighter than those provided by Qi (J. Symbolic Comput. 2005 ), Li et al. (Numer. Linear Algebra App. 2014), and Li and Li (Linear and Multilinear Algebra, 2015). Based on this set, we give an upper bound for the spectral radius of nonnegative tensors, and prove that the new bound is sharper than those provided by Yang and Yang (SIAM J. Matrix Anal. App. 2010) and Li et al. (Numer. Linear Algebra App. 2014).

New inequalities for the minimum eigenvalue of M-matrices

Some new inequalities for the minimum eigenvalue of M-matrices are established. These inequalities improve the results in [G. Tian and T. Huang, Inequalities for the minimum eigenvalue of M-matrices, Electr. J. Linear Algebra 20 (2010), pp. 291–302].

Geršgorin-type and Brauer-type eigenvalue localization sets of stochastic matrices

Shen et al. in [Some classes of nonsingular matrices with applications to localize the real eigenvalues of real matrices, Linear Algebra and its Applications, 447 (2014), pp. 74–87] modified the Geršgorin circle set to localize all eigenvalues different from for stochastic matrices using the least off-diagonal element of each column. However, the modification does not always work when the least off-diagonal element of each column is 0. In this paper, we provide new methods to modify the Geršgorin and Brauer eigenvalue localization set. And numerical examples are given to show that new eigenvalue localization sets capture all eigenvalues different from for stochastic matrices more precisely than those provided by Shen et al.

New practical criteria for ℋ-tensors and its application

Several new criteria for identifying -tensors are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are given. Advantages of the obtained results are illustrated by numerical examples.

Subdirect sums of Nekrasov matrices

Some sufficient conditions ensuring that the subdirect sum of Nekrasov matrices is in the class of Nekrasov matrices are given. In particular, we obtain that the -subdirect sum is a Nekrasov matrix, when is a Nekrasov matrix and is a strictly diagonally dominant matrix.

Subdirect sums of weakly chained diagonally dominant matrices

Some sufficient conditions ensuring that the subdirect sum of two weakly chained diagonally dominant matrices is in this class, are given. In particular, it is shown that the subdirect sum of overlapping principal submatrices of a weakly chained diagonally dominant matrix is also a weakly chained diagonally dominant matrix.

New regions including eigenvalues of Toeplitz matrices

Two new eigenvalue inclusion regions for matrices with a constant main diagonal are given. We then apply these results to Toeplitz matrices, and obtain two regions including all eigenvalues of Toeplitz matrices. Furthermore, it is proved that the new regions are tighter than those in [Melman A. Ovals of Cassini for Toeplitz matrices, Linear and Multilinear Algebra. 2012;60:189–199].

Singular value inclusion sets for rectangular tensors

Abstract Two singular value inclusion sets for rectangular tensors are given. These sets provide two upper bounds and lower bounds for the largest singular value of nonnegative rectangular tensors, which can be taken as a parameter of an algorithm presented by Zhou et al. (Linear Algebra Appl. 2013; 438: 959–968) such that the sequences produced by this algorithm converge rapidly to the largest singular value of an irreducible nonnegative rectangular tensor.

SDB-tensors and SQB-tensors

ABSTRACT In this paper, we propose four new classes of structured tensors: -tensors, -tensors, and prove that even order real symmetric SDB-tensors and SQB-tensors are positive definite, and that even order real symmetric -tensors and -tensors are positive semi-definite. These new classes of structured tensors provide two checkable sufficient conditions for the positive definite tensors and two checkable sufficient conditions for the positive semi-definite tensors.