ABSTRACT

In this paper, we give a note on the eigenvalue localization sets for tensors. We show that these sets are tighter than those provided by Li et al. (2014) [1] .

Keywords:

Tensor Eigenvalue, Localization Set, Tensor

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 $ℝ$ be the real field. An m-th order n dimensional square tensor $A$ consists of nm entries in $ℝ$ , which is defined as follows:

$A=\left(a_{i_1{i}_2\cdots i_m}\right),a_{i_1{i}_2\cdots i_m}\in ℝ,1\le i_1,i_2,\cdots i_m\le n.$

To an n-vector x, real or complex, we define the n-vector:

$A{x}^{m-1}={\left(\sum _{i_2,\cdots ,i_m=1}^n{a}_{i{i}_2\cdots i_m}{x}_{i_2}\cdots x_{i_m}\right)}_{1\le i\le n}.$

and

$x^{\left[m-1\right]}={\left(x_i^{m-1}\right)}_{1\le i\le n}\mathrm{.}$

If $A{x}^{m-1}=\lambda x^{\left[m-1\right]}$ , x and $\lambda$ are all real, then $\lambda$ is called an H-eigenvalue of $A$ and x an H-eigenvector of $A$ associated with $\lambda$ [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 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ be a complex tensor of order $m$ dimension $n$ . Then

$\sigma \left(A\right)\subseteq \Gamma \left(A\right)=\underset{i\in N}{\cup }{\Gamma }_i\left(A\right)$

where $\tau \left(A\right)$ is the set of all the eigenvalues of $A$ and

${\Gamma }_i\left(A\right)=\left\{z\in ℂ\mathrm{<mo>\; :\; </mo>}\left|z-a_{i\cdots i}\right|\le r_i\left(A\right)\right\}\mathrm{,}$

where

${\delta }_{i_1\cdots i_m}=\{\begin{array}{l}\mathrm{1,}\text{if}{i}_1=\cdots =i_m\\ \mathrm{0,}\text{otherwise}\mathrm{,}\end{array}$

and

$r_i\left(A\right)=\sum _{{\delta }_{i{i}_2\cdots i_m}=0}\left|a_{i{i}_2\cdots i_m}\right|\mathrm{.}$

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 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ be a complex tensor of order $m$ dimension $n$ . Then

$\sigma \left(A\right)\subseteq K\left(A\right)=\underset{i\mathrm{,}j\in N\mathrm{,}i\ne j}{\cup }{K}_{i\mathrm{,}j}\left(A\right)$

where $\sigma \left(A\right)$ is the set of all the eigenvalues of $A$ and

$K_{i\mathrm{,}j}\left(A\right)=\left\{z\in ℂ\mathrm{<mo>\; :\; </mo>}\left(\left|z-a_{i\cdots i}\right|-r_i^j\left(A\right)\right)\left|z-a_{j\cdots j}\right|\le \left|a_{ij\dots j}\right|r_j\left(A\right)\right\}\mathrm{,}$

where

$r_i^j\left(A\right)=\sum _{\begin{array}{l}{\delta }_{i{i}_2\cdots i_m}=0,\\ {\delta }_{j{i}_2\cdots i_m}=0\end{array}}\left|a_{i{i}_2\cdots i_m}\right|=r_i\left(A\right)-\left|a_{ij\cdots j}\right|.$

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 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ be a complex tensor of order $m$ dimension $n$ . Then

$\sigma \left(A\right)\subseteq \Delta \left(A\right)=\underset{i\in N}{\cap }\underset{j\in N\mathrm{,}j\ne i}{\cup }{\Delta }_{i\mathrm{,}j}\left(A\right)$

where $\sigma \left(A\right)$ is the set of all the eigenvalues of $A$ and

${\Delta }_{i\mathrm{,}j}\left(A\right)=\left\{z\in ℂ\mathrm{<mo>\; :\; </mo>}\left|z-a_{i\cdots i}\right|\left(\left|z-a_{j\cdots j}\right|-r_j^i\left(A\right)\right)\le \left|a_{ji\cdots i}\right|r_i\left(A\right)\right\}\mathrm{,}$

where

$r_j^i\left(A\right)=\sum _{\begin{array}{l}{\delta }_{j{i}_2\cdots i_m}=0,\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}\left|a_{j{i}_2\cdots i_m}\right|=r_j\left(A\right)-\left|a_{ji\cdots i}\right|\mathrm{.}$

Proof. Let $x={\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_n\right)}^{\text{T}}$ be an eigenvector of $A$ corresponding to $\lambda \left(A\right)$ , that is,

$A{x}^{m-1}=\lambda x^{\left[m-1\right]}\mathrm{.}$ (1)

Let

$\left|x_p\right|=max\left\{\left|x_i\right|\mathrm{,}i\in N\right\}\mathrm{.}$

Obviously, $\left|x_p\right|>0$ . For any $q\ne p$ , from equality (1), we have

$\begin{array}{c}\left|\lambda -a_{p\cdots p}\right|{\left|x_p\right|}^{m-1}\le \sum _{{\delta }_{p{i}_2\cdots i_m}=0}\left|a_{p{i}_2\cdots i_m}\right|\left|x_{i_2}\right|\cdots \left|x_{i_m}\right|\\ \le \sum _{\begin{array}{l}{\delta }_{q{i}_2\cdots i_m}=0,\\ {\delta }_{p{i}_2\cdots i_m}=0\end{array}}\left|a_{p{i}_2\cdots i_m}\right|\left|x_{i_2}\right|\cdots \left|x_{i_m}\right|+\left|a_{pq\cdots q}\right|{\left|x_q\right|}^{m-1}\\ \le \sum _{\begin{array}{l}{\delta }_{q{i}_2\cdots i_m}=0,\\ {\delta }_{p{i}_2\cdots i_m}=0\end{array}}\left|a_{p{i}_2\cdots i_m}\right|{\left|x_p\right|}^{m-1}+\left|a_{pq\cdots q}\right|{\left|x_q\right|}^{m-1}\\ \le r_p^q\left(A\right){\left|x_p\right|}^{m-1}+\left|a_{pq\cdots q}\right|{\left|x_q\right|}^{m-1}.\end{array}$ (2)

That is,

$\left(\left|\lambda -a_{p\cdots p}\right|-r_p^q\left(A\right)\right){\left|x_p\right|}^{m-1}\le \left|a_{pq\cdots q}\right|{\left|x_q\right|}^{m-1}\mathrm{.}$ (3)

If $\left|x_q\right|=0$ for all $q\ne p$ , then $\left|\lambda -a_{p\cdots p}\right|-r_p^q\left(A\right)\le 0$ , and $\lambda \in \Delta \left(A\right)$ . If $\left|x_q\right|>0$ , from equality (1), we have

$\left|\lambda -a_{q\cdots q}\right|{\left|x_q\right|}^{m-1}\le r_q\left(A\right){\left|x_p\right|}^{m-1}\mathrm{.}$ (4)

Multiplying inequalities (3) with (4), we have

$\left|\lambda -a_{q\cdots q}\right|\left(\left|\lambda -a_{p\dots p}\right|-r_p^q\left(A\right)\right)\le r_q\left(A\right)\left|a_{pq\cdots q}\right|\mathrm{,}$ (5)

which implies that $\lambda \in {\Delta }_{p\mathrm{,}q}\left(A\right)$ . From the arbitrariness of q, we have $\lambda \in \Delta \left(A\right)$ . ,

Remark 1. Obviously, we can get $K\left(A\right)\subseteq \Delta \left(A\right)$ . That is to say, our new eigenvalue inclusion sets are always tighter than the inclusion sets in Theorem 2.

Remark 2. If the tensor $A$ is nonnegative, from (5), we can get

$\left(\lambda -a_{q\cdots q}\right)\left(\lambda -a_{p\cdots p}-r_p^q\left(A\right)\right)\le r_q\left(A\right)a_{pq\cdots q}\mathrm{.}$

Then, we can get,

$\lambda \le \frac{1}{2}\left\{a_{p\cdots p}+a_{q\cdots q}+r_p^q\left(A\right)+{\Theta }_{p\mathrm{,}q}^{\frac{1}{2}}\left(A\right)\right\}$

where

${\Theta }_{p\mathrm{,}q}\left(A\right)={\left(a_{p\cdots p}-a_{q\cdots q}+r_p^q\left(A\right)\right)}^2+4a_{pq\cdots q}{r}_q\left(A\right)\mathrm{.}$

From the arbitrariness of q, we have

$\lambda \le \underset{i\in N}{max}\underset{j\in N\mathrm{,}j\ne i}{min}\frac{1}{2}\left\{a_{j\cdots j}+a_{i\cdots i}+r_j^i\left(A\right)+{\Theta }_{j\mathrm{,}i}^{\frac{1}{2}}\left(A\right)\right\}\mathrm{.}$

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.

Cite this paper

He, J., Liu, Y.M., Tian, J.K. and Liu, X.H. (2017) A Note on the Inclusion Sets for Tensors. Advances in Linear Algebra & Matrix Theory, 7, 67-71. https://doi.org/10.4236/alamt.2017.73006

References