A Note on the Inclusion Sets for Tensors

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

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

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:

$A=\left({a}_{{i}_{1}{i}_{2}\cdots {i}_{m}}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{{i}_{1}{i}_{2}\cdots {i}_{m}}\in ℝ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\le {i}_{1},{i}_{2},\cdots {i}_{m}\le n.$

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

$A{x}^{m-1}={\left(\underset{{i}_{2},\cdots ,{i}_{m}=1}{\overset{n}{\sum }}{a}_{i{i}_{2}\cdots {i}_{m}}{x}_{{i}_{2}}\cdots {x}_{{i}_{m}}\right)}_{1\le i\le n}.$

and

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

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

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

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

${\Gamma }_{i}\left(A\right)=\left\{z\in ℂ:|z-{a}_{i\cdots i}|\le {r}_{i}\left(A\right)\right\},$

where

${\delta }_{{i}_{1}\cdots {i}_{m}}=\left\{\begin{array}{l}1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{i}_{1}=\cdots ={i}_{m}\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{otherwise},\end{array}$

and

${r}_{i}\left(A\right)=\underset{{\delta }_{i{i}_{2}\cdots {i}_{m}}=0}{\sum }|{a}_{i{i}_{2}\cdots {i}_{m}}|.$

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

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

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

${K}_{i,j}\left(A\right)=\left\{z\in ℂ:\left(|z-{a}_{i\cdots i}|-{r}_{i}^{j}\left(A\right)\right)|z-{a}_{j\cdots j}|\le |{a}_{ij\dots j}|{r}_{j}\left(A\right)\right\},$

where

${r}_{i}^{j}\left(A\right)=\underset{\begin{array}{l}{\delta }_{i{i}_{2}\cdots {i}_{m}}=0,\\ {\delta }_{j{i}_{2}\cdots {i}_{m}}=0\end{array}}{\sum }|{a}_{i{i}_{2}\cdots {i}_{m}}|={r}_{i}\left(A\right)-|{a}_{ij\cdots j}|.$

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

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

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

${\Delta }_{i,j}\left(A\right)=\left\{z\in ℂ:|z-{a}_{i\cdots i}|\left(|z-{a}_{j\cdots j}|-{r}_{j}^{i}\left(A\right)\right)\le |{a}_{ji\cdots i}|{r}_{i}\left(A\right)\right\},$

where

${r}_{j}^{i}\left(A\right)=\underset{\begin{array}{l}{\delta }_{j{i}_{2}\cdots {i}_{m}}=0,\\ {\delta }_{i{i}_{2}\cdots {i}_{m}}=0\end{array}}{\sum }|{a}_{j{i}_{2}\cdots {i}_{m}}|={r}_{j}\left(A\right)-|{a}_{ji\cdots i}|.$

Proof. Let $x={\left({x}_{1},\cdots ,{x}_{n}\right)}^{\text{T}}$ be an eigenvector of $A$ corresponding to $\lambda \left(A\right)$ , that is,

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

Let

$|{x}_{p}|=max\left\{|{x}_{i}|,i\in N\right\}.$

Obviously, $|{x}_{p}|>0$ . For any $q\ne p$ , from equality (1), we have

$\begin{array}{c}|\lambda -{a}_{p\cdots p}|{|{x}_{p}|}^{m-1}\le \underset{{\delta }_{p{i}_{2}\cdots {i}_{m}}=0}{\sum }|{a}_{p{i}_{2}\cdots {i}_{m}}||{x}_{{i}_{2}}|\cdots |{x}_{{i}_{m}}|\\ \le \underset{\begin{array}{l}{\delta }_{q{i}_{2}\cdots {i}_{m}}=0,\\ {\delta }_{p{i}_{2}\cdots {i}_{m}}=0\end{array}}{\sum }|{a}_{p{i}_{2}\cdots {i}_{m}}||{x}_{{i}_{2}}|\cdots |{x}_{{i}_{m}}|+|{a}_{pq\cdots q}|{|{x}_{q}|}^{m-1}\\ \le \underset{\begin{array}{l}{\delta }_{q{i}_{2}\cdots {i}_{m}}=0,\\ {\delta }_{p{i}_{2}\cdots {i}_{m}}=0\end{array}}{\sum }|{a}_{p{i}_{2}\cdots {i}_{m}}|{|{x}_{p}|}^{m-1}+|{a}_{pq\cdots q}|{|{x}_{q}|}^{m-1}\\ \le {r}_{p}^{q}\left(A\right){|{x}_{p}|}^{m-1}+|{a}_{pq\cdots q}|{|{x}_{q}|}^{m-1}.\end{array}$ (2)

That is,

$\left(|\lambda -{a}_{p\cdots p}|-{r}_{p}^{q}\left(A\right)\right){|{x}_{p}|}^{m-1}\le |{a}_{pq\cdots q}|{|{x}_{q}|}^{m-1}.$ (3)

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

$|\lambda -{a}_{q\cdots q}|{|{x}_{q}|}^{m-1}\le {r}_{q}\left(A\right){|{x}_{p}|}^{m-1}.$ (4)

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

$|\lambda -{a}_{q\cdots q}|\left(|\lambda -{a}_{p\dots p}|-{r}_{p}^{q}\left(A\right)\right)\le {r}_{q}\left(A\right)|{a}_{pq\cdots q}|,$ (5)

which implies that $\lambda \in {\Delta }_{p,q}\left(A\right)$ . 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

$\left(\lambda -{a}_{q\cdots q}\right)\left(\lambda -{a}_{p\cdots p}-{r}_{p}^{q}\left(A\right)\right)\le {r}_{q}\left(A\right){a}_{pq\cdots q}.$

Then, we can get,

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

where

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

From the arbitrariness of q, we have

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

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.

Conflicts of Interest

The authors declare no conflicts of interest.

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