Abstract
In this paper, a new S-type eigenvalue localization set for a tensor is derived by dividing N={1,2,ldots,n} into disjoint subsets S and its complement. It is proved that this new set is sharper than those presented by Qi (J. Symb. Comput. 40:1302-1324, 2005), Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and Li et al. (Linear Algebra Appl. 481:36-53, 2015). As applications of the results, new bounds for the spectral radius of nonnegative tensors and the minimum H-eigenvalue of strong M-tensors are established, and we prove that these bounds are tighter than those obtained by Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and He and Huang (J. Inequal. Appl. 2014:114, 2014).
Highlights
1 Introduction Eigenvalue problems of higher order tensors have become an important topic in the applied mathematics branch of numerical multilinear algebra, and they have a wide range of practical applications, such as best-rank one approximation in data analysis [ ], higher order Markov chains [ ], molecular conformation [ ], and so forth
Some sets including all eigenvalues of tensors have been presented by some researchers [, – ]. If one of these sets for an even-order real symmetric tensor is in the right-half complex plane, we can conclude that the smallest H-eigenvalue is positive, the corresponding tensor is positive definite
As applications of the new S-type eigenvalue inclusion set, the other main results of this paper is to provide sharper bounds for the spectral radius of nonnegative tensors and the minimum H-eigenvalue of nonsingular M-tensors, which improve some existing ones
Summary
Eigenvalue problems of higher order tensors have become an important topic in the applied mathematics branch of numerical multilinear algebra, and they have a wide range of practical applications, such as best-rank one approximation in data analysis [ ], higher order Markov chains [ ], molecular conformation [ ], and so forth. As applications of the new S-type eigenvalue inclusion set, the other main results of this paper is to provide sharper bounds for the spectral radius of nonnegative tensors and the minimum H-eigenvalue of nonsingular M-tensors, which improve some existing ones. A new Stype eigenvalue inclusion set for tensors is given, and proved to be tighter than the existing ones derived in Lemmas . ([ ]) Let A be a strong M-tensor and denoted by τ (A) the minimum value of the real part of all eigenvalues of A. We establish a comparison theorem for the new S-type eigenvalue inclusion set derived in this paper and those in Lemmas . 3.2 A new upper bound for the spectral radius of nonnegative tensors Based on the results of Section .
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