1. Introduction
Eigenvalue problems of higher order tensors have become an important topic of study in a new applied mathematics branch, numerical multilinear algebra, and they have a wide range of practical applications [2] - [9] .
First, we recall some definitions on tensors. Let
be the real field. An m-th order n dimensional square tensor
consists of nm entries in
, which is defined as follows:
To an n-vector x, real or complex, we define the n-vector:
and
If
, x and
are all real, then
is called an H-eigenvalue of
and x an H-eigenvector of
associated with
[10] [11] .
Qi [10] generalized Geršgorin eigenvalue inclusion theorem from matrices to real supersymmetric tensors, which can be easily extended to generic tensors; see [1] .
Theorem 1. Let
be a complex tensor of order
dimension
. Then
where
is the set of all the eigenvalues of
and
where
and
Recently, Li et al. [1] obtained the following result, which is also used to identify the positive definiteness of an even-order real supersymmetric tensor.
Theorem 2. Let
be a complex tensor of order
dimension
. Then
where
is the set of all the eigenvalues of
and
where
In this paper, we give some new eigenvalue localization sets for tensors, which are tighter than those provided by Li et al. [1] .
2. New Eigenvalue Inclusion Sets
Theorem 3. Let
be a complex tensor of order
dimension
. Then
where
is the set of all the eigenvalues of
and
where
Proof. Let
be an eigenvector of
corresponding to
, that is,
(1)
Let
Obviously,
. For any
, from equality (1), we have
(2)
That is,
(3)
If
for all
, then
, and
. If
, from equality (1), we have
(4)
Multiplying inequalities (3) with (4), we have
(5)
which implies that
. From the arbitrariness of q, we have
. ,
Remark 1. Obviously, we can get
. That is to say, our new eigenvalue inclusion sets are always tighter than the inclusion sets in Theorem 2.
Remark 2. If the tensor
is nonnegative, from (5), we can get
Then, we can get,
where
From the arbitrariness of q, we have
That is to say, from Theorem 3, we can get another proof of the result in Theorem 13 in [12] .
Funds
Jun He is supported by Science and technology Foundation of Guizhou province (Qian ke he Ji Chu [2016]1161); Guizhou province natural science foundation in China (Qian Jiao He KY [2016]255); The doctoral scientific research foundation of Zunyi Normal College (BS [2015]09); High-level innovative talents of Guizhou Province (Zun Ke He Ren Cai [2017]8). Yan-Min Liu is supported by National Natural Science Foundations of China (71461027); Science and technology talent training object of Guizhou province outstanding youth (Qian ke he ren zi [2015]06); Guizhou province natural science foundation in China (Qian Jiao He KY [2014]295); 2013, 2014 and 2015 Zunyi 15,851 talents elite project funding; Zhunyi innovative talent team(Zunyi KH (2015)38). Tian is supported by Guizhou province natural science foundation in China (Qian Jiao He KY [2015]451); Scienceand technology Foundation of Guizhou province (Qian ke he J zi [2015]2147). Xiang-Hu Liu is supported by Guizhou Province Department of Education Fund KY [2015]391, [2016]046; Guizhou Province Department of Education teaching reform project [2015]337; Guizhou Province Science and technology fund (qian ke he ji chu) [2016]1160.