Show Summary Details
More options …

# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2017: 161.82

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 15, Issue 1

# Two new eigenvalue localization sets for tensors and theirs applications

Jianxing Zhao
• Corresponding author
• College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China
• Email
• Other articles by this author:
/ Caili Sang
Published Online: 2017-10-09 | DOI: https://doi.org/10.1515/math-2017-0106

## Abstract

A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324) and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50). As an application, a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1), 187-198). As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

MSC 2010: 15A18; 15A42; 15A69

## 1 Introduction

For a positive integer n, n ≥ 2, N denotes the set {1, 2, …, n}. ℂ(ℝ) denotes the set of all complex (real) numbers. We call 𝓐 = (ai1im) a complex (real) tensor of order m dimension n, denoted by 𝓐 ∈ ℂ[m,n](ℝ[m,n]), if $ai1⋯im∈C(R),$

where ijN for j = 1, 2, ⋯, m. A tensor of order m dimension n is called the unit tensor, denoted by 𝓘, if its entries are δi1im for i1, ⋯, imN, where $δi1⋯im=1,if i1=⋯=im,0,otherwise.$

𝓐 is called nonnegative if ai1im ≥ 0. 𝓐 = (ai1im) ∈ ℝ[m,n] is called symmetric [1] if $ai1⋯im=aπ(i1⋯im),∀π∈Πm,$

where Πm is the permutation group of m indices. 𝓐 = (ai1im) ∈ ℝ[m,n] is called weakly symmetric [2] if the associated homogeneous polynomial $Axm=∑i1,⋯,im∈Nai1⋯imxi1…xim$

satisfies ∇ 𝓐xm = m𝓐xm−1. It is shown in [2] that a symmetric tensor is necessarily weakly symmetric, but the converse is not true in general.

To an n-vector x = (x1, x2⋯, xn)T, real or complex, we define the n-vector: $Axm−1=∑i2,⋯,im∈Naii2⋯imxi2…xim1≤i≤n$

and $x[m−1]=(xim−1)1≤i≤n.$

#### Definition 1.1

([1, 3]). Let 𝓐 = (ai1im) ∈ ℂ[m,n]. A pair (λ, x) ∈ ℂ × (ℂn ∖ {0}) is called an eigenvalue-eigenvector (or simply eigenpair) of 𝓐 if $Axm−1=λx[m−1].$

(λ, x) is called an H-eigenpair if both of them are real.

#### Definition 1.2

([1, 3]). Let 𝓐 = (ai1im) ∈ ℝ[m,n]. A pair (λ, x) ∈ ℂ × (ℂn ∖ {0}) is called an E-eigenpair of 𝓐 if $Axm−1=λxandxTx=1.$

(λ, x) is called an Z-eigenpair if both of them are real.

We define the Z-spectrum of 𝓐, denoted 𝓩(𝓐) to be the set of all Z-eigenvalues of 𝓐. Assume 𝓩(𝓐) ≠ 0, then the Z-spectral radius [2] of 𝓐, denoted ϱ(𝓐), is defined as $ϱ(A):=sup{|λ|:λ∈Z(A)}.$

It is shown in [1] that a real even-order symmetric tensor 𝓐 = (ai1im) is positive definite if and only if all of its H-eigenvalues (Z-eigenvalues) are positive. However, when m and n are very large, it is not easy to compute all H-eigenvalues (Z-eigenvalues) of 𝓐. Then we can try to give a set in the complex which includes all H-eigenvalues (Z-eigenvalues) of 𝓐. If this set is in the right-half complex plane, then we can conclude that all H-eigenvalues (Z-eigenvalues) are positive, consequently, 𝓐 is positive definite; for details, see [1, 47].

There are other applications of (E-)eigenvalue inclusion sets, for example we can use them to obtain the lower and upper bounds for the H-eigenvalues (Z-spectral radius) of (nonnegative) tensors and the minimum eigenvalue of 𝓜-tensors; for details, see [819].

In 2005, Qi [1] presented the following Geršgorin-type eigenvalue localization set for real symmetric tensors, which can be easily extended to general tensors [4, 20].

#### Theorem 1.3

([1, Theorem 6]). Let 𝓐 = (ai1im) ∈ ℂ[m,n]. Then $σ(A)⊆Γ(A)=⋃i∈NΓi(A),$

where σ(𝓐) is the set of all eigenvalues of 𝓐 and $Γi(A)={z∈C:|z−ai⋯i|≤ri(A)},ri(A)=∑δii2⋯im=0|aii2⋯im|.$

To get a tighter eigenvalue localization set than Γ(𝓐), Li et al. [4] proposed the following Brauer-type eigenvalue localization set for tensors.

#### Theorem 1.4

([4, Theorem 2.1]). Let 𝓐 = (ai1im) ∈ ℂ[m,n]. Then $σ(A)⊆K(A)=⋃i,j∈N,j≠iKi,j(A),$

where $Ki,j(A)={z∈C:(|z−ai⋯i|−rij(A))|z−aj⋯j|≤|aij⋯j|rj(A)},rij(A)=∑δii2⋯im=0,δji2⋯im=0|aii2⋯im|=rj(A)−|aij⋯j|.$

To reduce computations, Li et al. [4] gave an S-type eigenvalue localization set by breaking N into disjoint subsets S and S, where S is the complement of S in N.

#### Theorem 1.5

([4, Theorem 2.2]). Let 𝓐 = (ai1im) ∈ ℂ[m,n], S be a nonempty proper subset of N. Then $σ(A)⊆KS(A)=⋃i∈S,j∈S¯Ki,j(A)⋃⋃i∈S¯,j∈SKi,j(A).$

In 2017, Wang et al. established the following Z-eigenvalue localization set for a real tensor 𝓐, which is completely different from eigenvalue localization sets and can be generalized to an E-eigenvalue localization set easily.

#### Theorem 1.6

([8, Theorem 3.1]). Let 𝓐 = (ai1im) ∈ ℝ[m,n]. Then $Z(A)⊆Γ^(A)=⋃i∈NΓ^i(A),$

where $Γ^i(A)={z∈C:|z|≤Ri(A)},Ri(A)=∑i2⋯im∈N|aii2⋯im|.$

The main aim of this paper is to give a new eigenvalue localization set for tensors, which is tighter than those in Theorems 1.3-1.5, and a new E-eigenvalue localization set for tensors, which is tighter than that in Theorem 1.6. As applications, a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor is obtained based on the eigenvalue localization set, and a new upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained based on the E-eigenvalue localization set.

## 2 A new eigenvalue localization set for tensors and its applications

In this section, we propose a new eigenvalue localization set for tensors and establish the comparisons between this set with those in Theorems 1.3-1.5. As an application of this set, we give a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor.

#### Theorem 2.1

Let 𝓐 = (ai1im) ∈ ℂ[m,n]. Then $σ(A)⊆K∩(A)=⋃i∈N⋂j∈N,j≠iKi,j(A),$

where $Ki,j(A)={z∈C:(|z−ai⋯i|−rij(A))|z−aj⋯j|≤|aij⋯j|rj(A)}.$

#### Proof

Let λ be an eigenvalue of 𝓐 with corresponding eigenvector x = (x1, ⋯, xn)T ∈ ℂn ∖ {0}, i.e., $Axm−1=λx[m−1].$(1)

Let |xp| = max{|xi| : iN}. Then, |xp| > 0. From (1), we have $(λ−ap⋯p)xpm−1=∑δpi2⋯im=0,δji2⋯im=0api2⋯jmxi2⋯xim+apj⋯jxjm−1,∀j∈N,j≠p.$

Taking modulus in the above equation and using the triangle inequality give $|λ−ap⋯p||xp|m−1≤∑δpi2⋯im=0,δji2⋯im=0|api2⋯im||xi2|⋯|xim|+|apj⋯j||xj|m−1≤∑δpi2⋯im=0,δji2⋯im=0|api2⋯im||xp|m−1+|apj⋯j||xj|m−1=rpj(A)|xp|m−1+|apj⋯j||xj|m−1,$

equivalently, $(|λ−ap⋯p|−rpj(A))|xp|m−1≤|apj⋯j||xj|m−1.$(2)

If |xj| = 0, by |xp| > 0, we have $\begin{array}{}|\lambda -{a}_{p\cdots p}|-{r}_{p}^{j}\left(\mathcal{A}\right)\le 0.\end{array}$ Then $(|λ−ap⋯p|−rpj(A))|λ−aj⋯j|≤0≤|apj⋯j|rj(A),$

which implies that λ ∈ 𝓚p,j(𝓐) ⊆ 𝓚(𝓐). otherwise, |xj| > 0. Similarly, from (1), we can obtain $|λ−aj⋯j||xj|m−1≤rj(A)|xp|m−1$(3)

Multiplying (2) with (3) and noting that |xp|m−1|xj|m−1 > 0, we have $(|λ−ap⋯p|−rpj(A))|λ−aj⋯j|≤|apj⋯j|rj(A),$

then λ ∈ 𝓚p,j(𝓐) ⊆ 𝓚(𝓐) . From the arbitrariness of j, we have $\begin{array}{}\lambda \in \bigcap _{j\in N,j\ne p}{\mathcal{K}}_{p,j}\left(\mathcal{A}\right).\end{array}$ Furthermore, λ$\begin{array}{}\bigcup _{i\in N}\bigcap _{j\in N,j\ne i}{\mathcal{K}}_{i,j}\left(\mathcal{A}\right).\end{array}$

Next, a comparison theorem is given for Theorems 1.3-1.5 and Theorem 2.1.

#### Theorem 2.2

Let 𝓐 = (ai1im) ∈ ℂ[m,n], S be a nonempty proper subset of N. Then $K∩(A)⊆KS(A)⊆K(A)⊆Γ(A).$

#### Proof

Let S be the complement of S in N. According to Theorem 2.3 in [4], 𝓚S(𝓐) ⊆ 𝓚(𝓐) ⊆Γ(𝓐) . Hence, we only prove 𝓚(𝓐) ⊆ 𝓚S(𝓐) . Let z ∈ 𝓚(𝓐), then there exists i0N, such that z ∈ 𝓚i0,j(𝓐), ∀ jN, ji0. If i0S, then for any jS, we have $\begin{array}{}z\in \bigcup _{{i}_{0}\in S,j\in \overline{S}}{\mathcal{K}}_{{i}_{0},j}\left(\mathcal{A}\right)\subseteq {\mathcal{K}}^{S}\left(\mathcal{A}\right).\end{array}$ If i0S, then for any jS, we have $\begin{array}{}z\in \bigcup _{{i}_{0}\in \overline{S},j\in S}{\mathcal{K}}_{i,j}\left(\mathcal{A}\right)\subseteq {\mathcal{K}}^{S}\left(\mathcal{A}\right).\end{array}$ The conclusion follows. □

#### Remark 2.3

Theorem 2.2 shows that this set in Theorem 2.1 is tighter than those in Theorem 1.3, Theorem 1.4 and Theorem 1.5, that is, 𝓚(𝓐) can capture all eigenvalues of 𝓐 more precisely than Γ(𝓐), 𝓚(𝓐) and 𝓚S(𝓐).

As shown in [1, 47], an eigenvalue localization set can provide a checkable sufficient condition for the positive (semi-)definiteness of tensors. As an application of Theorem 2.1, we give a checkable sufficient condition for the positive (semi-)definiteness of tensors.

#### Theorem 2.4

Let 𝓐 = (ai1im) ∈ ℝ[m,n] be an even-order symmetric tensor with akk > 0 for all kN. IfiN, ∃ jN, ji, $(ai⋯i−rij(A))aj⋯j>|aij⋯j|rj(A),$

then 𝓐 is positive definite.

#### Proof

Let λ be an H-eigenvalue of 𝓐. By Theorem 2.1, we have λ ∈ 𝓚(𝓐), that is, there is i0N, for any jN, ji0, $(|λ−ai0⋯i0|−ri0j(A))|λ−aj⋯j|≤|ai0j⋯j|rj(A).$

Suppose that λ ≤ 0. Then for i0N, ∃ j0, such that ai0i0 > 0, aj0j0 > 0, and $(|λ−ai0⋯i0|−ri0j0(A))|λ−aj0⋯j0|≥(ai0⋯i0−ri0j0(A))aj0⋯j0>|ai0j0⋯j0|rj0(A).$

This is a contradiction. Hence, λ > 0, and 𝓐 is positive definite. The conclusion follows. □

Similar to the proof of Theorem 2.4, the following sufficient condition is easily obtained.

#### Theorem 2.5

Let 𝓐 = (ai1im) ∈ ℝ[m,n] be an even-order symmetric tensor with akk ≥ 0 for all kN. IfiN, ∃ jN, ji, $(ai⋯i−rij(A))aj⋯j≥|aij⋯j|rj(A),$

then 𝓐 is positive semi-definite.

#### Remark 2.6

When n = 2, Theorem 2.4 is the same as Theorem 4.1 and Theorem 4.2 in [4]. When n ≥ 3, it is easy to see that the conditions of Theorem 2.4 for determining the positive definiteness of tensors are weaker than those in Theorem 4.1 and Theorem 4.2 in [4].

Next, an example is given to verify the fact in Remark 2.6.

#### Example 2.7

Let 𝓐 = (aijkl) ∈ ℝ[4,3] be a symmetric tensor with elements defined as follows: $a1111=12.1,a2222=4.6,a3333=3.6,a1112=−0.1,a1113=0.15,a1122=−0.2,a1123=−0.2,a1133=0,a1222=−0.1,a1223=0.3,a1233=0.1,a1333=−0.15,a2223=0.1,a2233=−0.1,a2333=0.2.$

By computations, we get that $(a1111−r13(A))a3333=29.7>0.5550=|a1333|r3(A);$(4) $(a2222−r23(A))a3333=1.0800>0.7400=|a2333|r3(A);$(5) $(a3333−r31(A))a1111=0.6050>0.6000=|a3111|r1(A);$(6) $(a3333−r32(A))a2222=0<0.45=|a3222|r2(A).$(7)

Let S = {1, 2}, S = {3}. Because (7) holds, we can not use Theorem 4.1 and Theorem 4.2 in [4] to determine the positiveness of 𝓐 under this division. But from (4)-(6) and Theorem 2.4, we can determine that 𝓐 is positive definite. In fact, all the H-eigenvalues of 𝓐 are 2.9074, 3.1633, 3.7705, 4.6282 and 12.4216. By Theorem 5 in [1], 𝓐 is positive definite.

## 3 A new E-eigenvalue localization set for tensors and its applications

In this section, we give an S-type E-eigenvalue localization set for tensors, and establish the comparison between this set with that in Theorem 1.6. For simplification, we first denote some notations. Given a nonempty proper subset S of N, let $ΔN={(i2,i3,⋯,im):eachij∈Nforj=2,⋯,m},ΔS={(i2,i3,⋯,im):eachij∈Sforj=2,⋯,m},$

and then $ΔS¯=ΔN∖ΔS.$

This implies that for a tensor 𝓐 = (ai1im) ∈ ℝ[m,n], we have that for iS, $Ri(A)=∑i2,⋯,im∈N|aii2⋯im|=RiΔS(A)+RiΔS¯(A),$

where $RiΔS(A)=∑(i2,⋯,im)∈ΔS|aii2⋯im|,RiΔS¯(A)=∑(i2,⋯,im)∈ΔS¯|aii2⋯im|.$

#### Theorem 3.1

Let 𝓐 = (ai1im) ∈ ℝ[m,n], S be a nonempty proper subset of N, S be the complement of S in N. Then $σE(A)⊆ΩS(A)=⋃i∈S,j∈S¯Ωi,jS(A)⋃⋃i∈S¯,j∈SΩi,jS¯(A),$

where σE(𝓐) is the set of all E-eigenvalues of 𝓐 and $Ωi,jS(A)={z∈C:|z|(|z|−RjΔS¯(A))≤Ri(A)RjΔS¯(A)},Ωi,jS¯(A)={z∈C:|z|(|z|−RjΔS¯¯(A))≤Ri(A)RjΔS¯(A)}.$

#### Proof

Let λ be an E-eigenvalue of 𝓐 with corresponding eigenvector x = (x1,⋯,xn)T ∈ ℂn ∖ {0}, i.e., $Axm−1=λx,||x||2=1.$(8)

Let |xp| = max{|xi| : iS} and |xq| = max{|xj| : jS}. Then, at least one of |xp| and |xq| is nonzero. We next distinguish two cases to prove.

Case I. Suppose that $\begin{array}{}|{x}_{q}|\ge |{x}_{p}|,\text{\hspace{0.17em}then\hspace{0.17em}}|{x}_{q}|=\underset{j\in N}{max}|{x}_{j}|\text{\hspace{0.17em}and\hspace{0.17em}}0<|{x}_{q}{|}^{m-1}\le |{x}_{q}|\le 1.\end{array}$ From (8), we have $λxq=∑(i2⋯im)∈ΔSaqi2⋯imxj2⋯xim+∑(i2⋯im)∈ΔS¯aqi2⋯imxi2⋯xim.$

Taking modulus in the above equation and using the triangle inequality give $|λ||xq|m−1≤|λ||xq|≤∑(i2⋯im)∈ΔS|aqi2⋯im||xi2|⋯|xim|+∑(i2⋯im)∈ΔS¯|aqi2⋯im||xi2|⋯|xim|≤∑(i2⋯im)∈ΔS|aqi2⋯im||xp|m−1+∑(i2⋯im)∈ΔS¯|aqi2⋯im||xq|m−1=RqΔS(A)|xp|m−1+RqΔS¯(A)|xq|m−1,$

i.e., $(|λ|−RqΔS¯(A))|xq|m−1≤RqΔS(A)|xp|m−1.$(9)

If |xp| = 0, by |xq| > 0, we have $\begin{array}{}|\lambda |-{R}_{q}^{\overline{{\mathrm{\Delta }}^{S}}}\left(\mathcal{A}\right)\le 0.\end{array}$ Then $(|λ|−RqΔS¯(A))|λ|≤0≤RqΔS(A)Rp(A),$

which implies that $\begin{array}{}\lambda \in {\mathrm{\Omega }}_{p,q}^{S}\left(\mathcal{A}\right)\subseteq {\mathrm{\Omega }}^{S}\left(\mathcal{A}\right).\end{array}$ If |xp| > 0, from (8), we can obtain $|λ||xp|m−1≤|λ||xp|≤∑i2⋯im∈N|api2⋯im||xi2|⋯|xim|≤Rp(A)|xq|m−1.$(10)

Multiplying (9) with (10) and noting that |xp|m−1|xq|m−1 > 0, we have $(|λ|−RqΔS¯(A))|λ|≤RqΔS(A)Rp(A),$

which leads to $\begin{array}{}\lambda \in {\mathrm{\Omega }}_{p,q}^{S}\left(\mathcal{A}\right)\subseteq {\mathrm{\Omega }}^{S}\left(\mathcal{A}\right).\end{array}$

Case II. Suppose that |xp| ≥ |xq|, then $\begin{array}{}|{x}_{p}|=\underset{i\in N}{max}|{x}_{i}|\end{array}$ and 0 < |xp|m−1 ≤ |xp| ≤ 1. Similar to (9), we can obtain $(|λ|−RpΔs¯¯(A))|xp|m−1≤RpΔS¯(A)|xq|m−1.$(11)

If |xq| = 0, by |xp| > 0, we have $\begin{array}{}|\lambda |-{R}_{p}^{\overline{\mathrm{\Delta }\overline{s}}}\left(\mathcal{A}\right)\le 0.\end{array}$ Then $(|λ|−RpΔS¯¯(A))|λ|≤0≤RpΔS¯(A)Rq(A),$

which implies that $\begin{array}{}\lambda \in {\mathrm{\Omega }}_{q,p}^{\overline{S}}\left(\mathcal{A}\right)\subseteq {\mathrm{\Omega }}^{S}\left(\mathcal{A}\right).\end{array}$ If |xq| > 0, similar to (10), we have $|λ||xq|m−1≤Rq(A)|xp|m−1.$(12)

Multiplying (11) with (12) and noting that |xp|m−1|xq|m−1 > 0, we have $(|λ|−RpΔS¯¯(A))|λ|≤RpΔS¯(A)Rq(A),$

which leads to $\begin{array}{}\lambda \in {\mathrm{\Omega }}_{q,p}^{\overline{S}}\left(\mathcal{A}\right)\subseteq {\mathrm{\Omega }}^{\overline{S}}\left(\mathcal{A}\right).\end{array}$ The conclusion follows from Cases I and II. □

#### Theorem 3.2

Let 𝓐 = (ai1im) ∈ ℝ[m,n], S be a nonempty proper subset of N, S be the complement of S in N. Then $ΩS(A)⊆Γ^(A).$

#### Proof

Let λ ∈ ΩS(𝓐) . Then $λ∈⋃i∈S,j∈S¯Ωi,jS(A) or λ∈⋃i∈S¯,j∈SΩi,jS¯(A).$

Without loss of generality, suppose that $\begin{array}{}\lambda \in \bigcup _{i\in S,j\in \overline{S}}{\mathrm{\Omega }}_{i,j}^{S}\left(\mathcal{A}\right)\end{array}$ (we can prove it similarly if $\begin{array}{}\lambda \in \bigcup _{i\in \overline{S},j\in S}{\mathrm{\Omega }}_{i,j}^{\overline{S}}\left(\mathcal{A}\right)\right).\end{array}$

Then there are iS and jS such that $\begin{array}{}\lambda \in {\mathrm{\Omega }}_{i,j}^{S}\left(\mathcal{A}\right)\end{array}$, i.e., $|λ|(|λ|−RjΔS¯(A))≤Ri(A)RjΔS(A).$(13)

If $\begin{array}{}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}Rj\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(\mathcal{A}\right){R}_{j}^{{\mathrm{\Delta }}^{S}}\left(\mathcal{A}\right)=0,\text{\hspace{0.17em}then\hspace{0.17em}}\lambda =0\text{\hspace{0.17em}or\hspace{0.17em}}|\lambda |\le {R}_{j}^{\overline{{\mathrm{\Delta }}^{S}}}\left(\mathcal{A}\right)\le \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}Rj\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(\mathcal{A}\right).\end{array}$ Hence, λRi (𝓐) ⋃ Rj(𝓐). otherwise, from (13), we have $|λ|Ri(A)|λ|−RjΔS¯(A)RjΔS(A)≤1.$

Furthermore, $|λ|Ri(A)≤1$

or $|λ|−RjΔS¯(A)RjΔS(A)≤1,$

which implies that λRi (𝓐)⋃ Rj(𝓐) . □

#### Remark 3.3

From Theorem 3.2, we known that the set ΩS(𝓐) in Theorem 3.1 localizes all E-eigenvalues of a tensor 𝓐 more precisely than the set $\begin{array}{}\stackrel{^}{\mathrm{\Gamma }}\left(\mathcal{A}\right)\end{array}$ in Theorem 1.6.

Next, based on Theorem 3.1, we give an S-type upper bound for the Z-spectral radius of a weakly symmetric nonnegative tensor.

#### Theorem 3.4

Let 𝓐 = (ai1im) ∈ ℝ[m,n] be a weakly symmetric nonnegative tensor, S be a nonempty proper subset of N, S be the complement of S in N. Then $ϱ(A)≤ΨS(A)=maxmaxi∈S,j∈S¯ΨijS(A),maxi∈S,j∈S¯ΨijS¯(A),$

where $ΨijS(A)=12RjΔS¯(A)+[(RjΔS¯)2+4Ri(A)RjΔS(A)]12,ΨijS¯(A)=12RjΔS¯¯(A)+[(RjΔS¯¯)2+4Ri(A)RjΔS¯(A)]12.$

#### Proof

By Lemma 4.4 in [8], ϱ(𝓐) is the largest Z-eigenvalue of 𝓐. From Theorem 3.1, we know that ϱ(𝓐) ∈ ΩS(𝓐) . Then $ϱ(A)∈⋃i∈S,j∈S¯Ωi,jS(A) or ϱ(A)∈⋃i∈S¯,j∈SΩi,jS¯(A).$

We next distinguish two cases to prove.

Case I: If $\begin{array}{}\varrho \left(\mathcal{A}\right)\in \bigcup _{i\in S,j\in \overline{S}}{\mathrm{\Omega }}_{i,j}^{S}\left(\mathcal{A}\right),\end{array}$ then there exists i ∈ S, jS, such that $ϱ(A)(ϱ(A)−RjΔS¯(A))≤Ri(A)RjΔS(A).$

Then $ϱ(A)≤12RjΔS¯(A)+[(RjΔS¯(A))2+4Ri(A)RjΔS(A)]12.$

Furthermore, $ϱ(A)≤maxi∈S,j∈S¯12RjΔS¯(A)+[(RjΔS¯(A))2+4Ri(A)RjΔS(A)]12.$

Case II: If $\begin{array}{}\varrho \left(\mathcal{A}\right)\in \bigcup _{i\in \overline{S},j\in S}{\mathrm{\Omega }}_{i,j}^{\overline{S}}\left(\mathcal{A}\right),\end{array}$ similar to the proof of Case I, we can obtain $ϱ(A)≤maxi∈S¯,j∈S12RjΔS¯¯(A)+[(RjΔS¯¯(A))2+4Ri(A)RjΔS¯(A)]12.$

The conclusion follows from Cases I and II. □

#### Theorem 3.5

Let 𝓐 = (ai1im) ∈ ℝ[m,n] be a weakly symmetric nonnegative tensor, S be a nonempty proper subset of N, S be the complement of S in N. Then $ΨS(A)≤maxi∈NRi(A).$(14)

#### Proof

Here, we only prove that when $\begin{array}{}{\mathrm{\Psi }}^{S}\left(\mathcal{A}\right)=\underset{i\in S,j\in \overline{S}}{max}{\mathrm{\Psi }}_{ij}^{S}\left(\mathcal{A}\right),\end{array}$ (14) holds. Similarly, we can also prove that (14) holds if $\begin{array}{}{\mathrm{\Psi }}^{S}\left(\mathcal{A}\right)=\underset{i\in \overline{S},j\in S}{max}{\mathrm{\Psi }}_{ij}^{\overline{S}}\left(\mathcal{A}\right).\end{array}$ Next, we divide two cases to prove.

Case I: For any iS,jS, if Ri(𝓐) ≤ Rj(𝓐), then $ΨS(A)=maxi∈S,j∈S¯12RjΔS¯(A)+[(RjΔS¯(A))2+4Ri(A)RjΔS(A)]12≤maxi∈S,j∈S¯12RjΔS¯(A)+[(RjΔS¯(A))2+4Rj(A)RjΔS(A)]12=maxi∈S,j∈S¯12RjΔS¯(A)+[(RjΔS¯(A))2+4(RjΔS¯(A)+RjΔS(A))RjΔS(A)]12=maxi∈S,j∈S¯12RjΔS¯(A)+[(RjΔS¯(A)+2RjΔS(A))2]12=maxi∈S,j∈S¯12RjΔS¯(A)+RjΔS¯(A)+2RjΔS(A)=maxj∈S¯Rj(A).$

Case II: For any iS,jS, if Rj(𝓐) ≥ Ri(𝓐), then 0 ≤ $\begin{array}{}{R}_{j}^{{\mathrm{\Delta }}^{S}}\left(\mathcal{A}\right)\le {R}_{i}\left(\mathcal{A}\right)-{R}_{j}^{\overline{{\mathrm{\Delta }}^{S}}}\left(\mathcal{A}\right),\end{array}$ and $ΨS(A)=maxi∈S,j∈S¯12RjΔS¯(A)+[(RjΔS¯(A))2+4Ri(A)RjΔS(A)]12≤maxi∈S,j∈S¯12RjΔS¯(A)+[(RjΔS¯(A))2+4Ri(A)(Ri(A)−RjΔS¯(A))]12=maxi∈S,j∈S¯12RjΔS¯(A)+[(2Ri(A)−RjΔS¯(A))2]12=maxi∈S,j∈S¯12RjΔS¯(A)+2Ri(A)−RjΔS¯(A)=maxi∈SRi(A)≤maxi∈NRi(A).$

The conclusion follows from Cases I and II. □

#### Remark 3.6

Theorem 3.5 shows that the upper bound in Theorem 3.4 is better than Corollary 4.5 of [9].

Now, we show that the upper bound in Theorem 3.4 is sharper than those in [813] in some cases by the following example.

#### Example 3.7

Let 𝓐 = (aijk) ∈ ℝ[3,3] with entries defined as follows: $A(:,:,1)=0201.512020,A(:,:,2)=10.5200222.53,A(:,:,3)=020.5232.5121.$

It is not difficult to verify that 𝓐 is a weakly symmetric nonnegative tensor By computation, we obtain (ϱ(𝓐),x) = (7.3450, (0.3908, 0.6421, 0.6596)). By Corollary 4.5 of [9] and Theorem 3.3 of [10], we both have $ϱ(A)≤14.$

By Theorem 3.5 of [11], we have $ϱ(A)≤13.9189.$

By Theorem 4.6 of [8], we have $ϱ(A)≤13.9133.$

By Theorem 4.7 of [8], we have $ϱ(A)≤13.8167.$

By Theorem 4.5 of [8] and Theorem 6 of [12], we both have $ϱ(A)≤13.5000.$

By Theorem 2.9 of [13], we have $ϱ(A)≤12.9790.$

Let S = {1},S = {2,3}. By Theorem 3.4, we obtain $ϱ(A)≤11.5440.$

which shows that the upper bound in Theorem 3.4 is sharper.

## 4 Conclusions

In this paper, we give a new eigenvalue localization set 𝓚(𝓐) and prove that 𝓚(𝓐) is tighter than those in [1] and [4]. Based on this set, we obtain a weaker checkable sufficient condition to determine the positive (semi-)definiteness for an even-order real symmetric tensor. Meanwhile, we present an S-type E-eigenvalue localization set ΩS(𝓐) and prove that ΩS(𝓐) is tighter than that in [8]. As an application, we obtain an S-type upper bound ΨS(𝓐) for the Z-spectral radius of weakly symmetric nonnegative tensors, and show that ΨS(𝓐) is sharper than those in [813] in some cases by a numerical example. Then an interesting problem is how to pick S to make ΨS(𝓐) as small as possible. But this is difficult when n is large. In the future, we will focus on this problem.

## Acknowledgement

The authors are very indebted to the reviewers for their valuable comments and corrections, which improved the original manuscript of this paper. This work is supported by National Natural Science Foundation of China (No.11501141), Foundation of Guizhou Science and Technology Department (Grant No.[2015]2073) and Natural Science Programs of Education Department of Guizhou Province (Grant No.[2016]066).

## References

• [1]

Qi L.Q., Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 2005, 40, 1302–1324

• [2]

Chang K.C., Qi L.Q., Zhang T., A survey on the spectral theory of nonnegative tensors, Numer. Linear Algebra Appl., 2013, 20, 891–912

• [3]

Lim L.H., Singular values and eigenvalues of tensors: A variational approach, Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP’05), 2015, 1, 129–132 Google Scholar

• [4]

Li C.Q., Li Y.T., Kong X., New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 2014, 21, 39–50

• [5]

Li C.Q., Li Y.T., An eigenvalue localizatiom set for tensor with applications to determine the positive (semi-)definitenss of tensors, Linear Multilinear Algebra, 2016, 64(4), 587–601

• [6]

Li C.Q., Jiao A.Q., Li Y.T., An S-type eigenvalue location set for tensors, Linear Algebra Appl., 2016, 493, 469–483

• [7]

Li C.Q., Chen Z., Li Y.T., A new eigenvalue inclusion set for tensors and its applications, Linear Algebra Appl., 2015, 481, 36–53

• [8]

Wang G., Zhou G.L., Caccetta L., Z-eigenvalue inclusion theorems for tensors, Discrete Cont. Dyn.-B, 2017, 22(1), 187–198 Google Scholar

• [9]

Song Y.S., Qi L.Q., Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM. J. Matrix Anal. Appl., 2013, 34, 1581–1595

• [10]

Li W., Liu D.D., Vong S.W., Z-eigenvalue bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 2015, 483, 182–199

• [11]

He J., Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 2016, 20(7), 1290–1301 Google Scholar

• [12]

He J., Liu Y.M., Ke H., Tian J.K., Li X., Bounds for the Z-spectral radius of nonnegative tensors. Springerplus, 2016, 5(1), 1727

• [13]

Liu Q.L., Li Y.T., Bounds for the Z-eigenpair of general nonnegative tensors, Open Mathematics, 2016, 14, 181–194

• [14]

Huang Z.G., Wang L.G., Xu Z., Cui J.J., A new S-type eigenvalue inclusion set for tensors and its applications, J. Inequ. Appl., 2016, 2016, 254

• [15]

He J., Huang T.Z., Inequalities for 𝓜-tensors, J. Inequal. Appl., 2014, 2014, 114

• [16]

Zhao J.X., Sang C.L., Two new lower bounds for the minimum eigenvalue of 𝓜-tensors, J. Inequal. Appl., 2016, 2016, 268

• [17]

Zhou D.M., Chen G.L., Wu G.X., Zhang X.Y., On some new bounds for eigenvalues of the Hadamard product and the Fan product of matrices, Linear Algebra Appl., 2013, 438, 1415–1426

• [18]

Zhou D.M., Chen G.L., Wu G.X., Zhang X.Y., Some inequalities for the Hadamard product of an M-matrix and an inverse M-matrix, J. Inequal. Appl., 2013, 2013, 16

• [19]

Jin H.W., Kannan M.R., Bai M.R., Lower and upper bounds for the H-eigenvalues of even order real symmetric tensors, Linear Multilinear Algebra, 2017, 65(7), 1402–1416

• [20]

Yang Y.N., Yang Q.Z., Further results for Perron-Frobenius Theorem for nonnegative tensors, SIAM. J. Matrix Anal. Appl., 2010, 31, 2517–2530

Accepted: 2017-08-29

Published Online: 2017-10-09

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1267–1276, ISSN (Online) 2391-5455,

Export Citation

## Citing Articles

[1]
Caili Sang and Chaoqian Li
Linear and Multilinear Algebra, 2018, Page 1