Exclusion sets in the S-type eigenvalue localization sets for tensors

Yuan Zhang; Ying Zhang; Gang Wang

doi:10.1515/math-2019-0090

Article Open Access

Exclusion sets in the S-type eigenvalue localization sets for tensors

Yuan Zhang , Ying Zhang and Gang Wang

Published/Copyright: October 13, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this paper, we break the index set N into disjoint subsets S and its complement, and propose two S-type exclusion sets that all the eigenvalues do not belong to them. Furthermore, we establish new S-type eigenvalue inclusion sets, which can reduce computations and obtain more accurate numerical results. At the same time, we give two criteria for identifying nonsingular tensors. Finally, new S-type eigenvalue inclusion sets are shown to be sharper than existing results via two examples.

Keywords: S-type eigenvalue inclusion sets; S-type eigenvalue exclusion sets; Brauer-type eigenvalue inclusion sets

MSC 2010: 15A18; 15A42

1 Introduction

Let ℂ (ℝ) be the set of all complex (real) numbers, ℝ₊ (ℝ₊₊) be the set of all nonnegative (positive) numbers, ℂⁿ (ℝⁿ) be the set of all dimension n complex (real) vectors, and R+n(R++n) be the set of all dimension n nonnegative (positive) vectors. An m order n dimensional tensor (𝓐 = (a_{i₁i₂⋯i_m}) is a higher-order generalization of matrices, which consists of (n^m) entries:

ai1i2⋯im∈R,ik∈N={1,2,…,n},k=1,2,…,m.

𝓐 is called nonnegative (positive) if a_{i₁i₂…i_m} ∈ ℝ₊ (a_{i₁i₂…i_m} ∈ ℝ₊₊).

Generally, tensor is a higher-order extension of matrix, and hence many concepts and the corresponding conclusions for matrices such as determinant, eigenvalue and singular value theory are extended to higher order tensors by studying their multilinear algebra properties [1, 2]. Based on matrix eigenvalues, tensor eigenvalue problems are developed [3, 4] which are important tools in tensor analysis and computing, and they are widely used in medical resonance imaging [5], data analysis [6], higher-order Markov chains [7, 8] and positive definiteness of even-order multivariate forms in automatical control [9, 10, 11]. At the same time, many effective algorithms for finding the eigenvalue of tensors have been presented; for more detailed discussions, see [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. However, these algorithms cannot capture all eigenvalues, and their computational complexities are NP-hard. Hence, several inclusion sets in the complex plane for all the eigenvalues of a tensors have been considered: Gersgorin inclusion sets [4], Brauer inclusion sets [23, 24, 25, 26, 27, 28, 29, 30, 31], Brualdi inclusion sets [32] and the minimal Gersgorin sets [33]. For example, Gersgorin inclusion sets can be described as follows:

Lemma 1.1

(Theorem 6 of [4]) Let 𝓐 be a complex tensor of order m and dimension n. Then, all eigenvalues of 𝓐 are located in the union of the following sets:

σ(A)⊆Γ(A)=⋃i∈NΓi(A).

where σ(𝓐) denotes the set of all the set of all eigenvalues of 𝓐, Γ_i(𝓐) = {z ∈ 𝓒 : |z − a_i…i| ≤ r_i(𝓐)}, and ri(A)=∑δii2…im=0|aii2…im|.

On the other hand, by the discreteness of the eigenvalues [4], we know that all the eigenvalues are excluded some open sets. For this, Li et al. [34] proposed some exclusion sets, which exclude open sets respectively from the Gersgorin eigenvalue inclusion set in [4] and the Brauer-type eigenvalue inclusion set in [23] as follows:

Lemma 1.2

(Theorem 2 of [34]) Let 𝓐 be a complex tensor of order m and dimension n ≥ 2. Then, all eigenvalues of 𝓐 are located in the union of the following sets:

σ(A)⊆Ω(A)=⋃i∈NΩi(A).

where Ω_i(𝓐) = Γ_i(𝓐) \ Δ_i (𝓐), Δ_i (𝓐) = ⋃j≠i Δ_ij (𝓐) and

Δij(A)={z∈C:|z−aj…j|<2|aji…i|−rj(A)}.

Lemma 1.3

(Theorem 4 of [24]) Let 𝓐 be a complex tensor of order m and dimension n ≥ 2. Then, all eigenvalues of 𝓐 are located in the union of the following sets:

σ(A)⊆Θ(A)=⋃i,j∈N,j≠iΘij(A),

where Θ_ij(𝓐) = 𝓚_i,j(𝓐) ∖ Λ_i (𝓐), Λ_i(𝓐) = ⋃j≠i Λ_ij (𝓐) and

Ki,j(A)={z∈C:(|z−ai…i|−(ri(A)−|aij…j|))|z−aj…j|≤|aij…j|rj(A)},Λij(A)={z∈C:(|z−ai…i|+(ri(A)−|aij…j|))|z−aj…j|<|aij…j|(2|aji…i|−rj(A))}.

From Lemmas 1.1-1.3, we obtain inclusion sets Γ(𝓐) and 𝓚(𝓐) by computing n sets Γ_i(𝓐) and n(n − 1) sets 𝓚_i,j(𝓐), respectively. It is noted that 𝓚(𝓐) is much sharper than Γ(𝓐) [23]. Similarly, the exclusion sets Δ(𝓐) and Λ(𝓐) are composed of n(n − 1) sets Δ_i,j(𝓐) and n(n − 1) sets Λ_ij(𝓐), respectively. To reduce computations, Li et al. [23] gave an S-type eigenvalue inclusion set by selecting subset S of N, which 𝓚^S(𝓐) is made up of 2|S|(n − |S|) sets 𝓚_i,j(𝓐).

In this paper, based on S-type eigenvalue inclusion sets [23, 24], we propose S-type eigenvalue exclusion sets, which can reduce computations and achieve more accurate numerical results. Furthermore, we establish new S-type eigenvalue inclusion sets and propose criteria for identifying nonsingular tensors. To end this section, we introduce some fundamental notion and properties related to eigenvalue of a tensor [2, 3, 4] and propose structure of the article.

Let 𝓐 be an m-order n-dimensional tensor. Assume that 𝓐x^m−1 is not identical to 0. We say that (λ, x) ∈ 𝓒 × (𝓒ⁿ ∖ {0}) is an eigenvalue-eigenvector of 𝓐 if

Axm−1=λx[m−1],

where (Axm−1)i=∑i2,…,im=1naii2…imxi2…xim, x[m−1]=[x1m−1,x2m−1,…,xnm−1]T, and (λ, x) is called an H-eigenpair if they are both real.

We define the following m-order Kronecker delta

δi1i2…im=1, if i1=i2=⋯=im,0, otherwise.

This paper is organized as follows. In Section 1, we introduce important notation and recall fundamental results. In Section 2, we propose two S-type exclusion sets and establish new S-type eigenvalue inclusion sets. Meanwhile, we give two sufficient conditions to verify whether the determinant of a tensor is zero. In Section 3, we show that two new S-type eigenvalue inclusion sets are sharper than existing results via two examples.

2 S-type eigenvalue exclusion sets for tensors

In this section, we present two new S-type eigenvalue exclusion sets and show that these exclusion sets are included S-type inclusion set in [23, 24].

Lemma 2.1

(Theorem 2.2 of [23]) Let 𝓐 be a complex tensor of order m and dimension n ≥ 2 and S be a nonempty proper subset of N. Then, all eigenvalues of 𝓐 are located in the union of the following sets:

σ(A)⊆KS(A)=(⋃i∈S,j∈S¯Ki,j(A))⋃(⋃i∈S¯,j∈SKi,j(A)),

where 𝓚_i,j(𝓐) is defined in Lemma 1.3, S̄ is the complement of S in N.

Next we try to find some proper subsets of 𝓚^S(𝓐) which do not include any eigenvalue of a tensor 𝓐.

Theorem 2.2

Let 𝓐 be a complex tensor of order m and dimension n ≥ 2 and S be a nonempty proper subset of N. Then, all eigenvalues of 𝓐 are located in the union of the following sets:

σ(A)⊆ΨS(A)=(⋃i∈S,j∈S¯Ψi,j(A))⋃(⋃i∈S¯,j∈SΨi,j(A)),

where Ψ_i,j(𝓐) = 𝓚_i,j(𝓐) ∖ 𝓤_i,j(𝓐) and

Ui,j(A)={z∈C:(|z−ai…i|+ri(A)−|aij…j|)|z−aj…j|<|aij…j|(2|aji…i|−rj(A))}.

Furthermore, Ψ^S(𝓐) ⊆ 𝓚^S(𝓐).

Proof

Let λ be an eigenvalue of 𝓐 with corresponding eigenvector x, i.e.,

Axm−1=λx[m−1]. (1)

Let |x_t| = max {|x_i| : i ∈ S} and |x_p| = max {|x_i| : i ∈ S̄}. Then at least one of |x_t| and |x_p| is nonzero. We break the proof into three cases.

If |x_t||x_p| ≠ 0 and |x_t| ≥ |x_p|, then |x_t| = max {|x_i| : i ∈ N} ≠ 0. By (1), it holds that

(λ−ap…p)xpm−1=∑δti2…im=0δpi2…im=0api2…imxi2…xim+apt…txtm−1

and

apt…txtm−1=(λ−ap…p)xpm−1−∑δti2…im=0δpi2…im=0api2…imxi2…xim.

Taking modulus in the above equation and using the triangle inequality give

|apt…t||xt|m−1≤|λ−ap…p||xp|m−1+∑δti2…im=0δpi2…im=0|api2…im||xt|m−1=|λ−ap…p||xp|m−1+(rp(A)−|apt…t|)|xt|m−1,

equivalently,

(2|apt…t|−rp(A))|xt|m−1≤|λ−ap…p||xp|m−1. (2)

Similarly, by the t-th equation of (1), we obtain

(λ−at…t)xtm−1=∑δti2…im=0δpi2…im=0ati2…imxi2…xim+atp…pxpm−1

and

|atp…p||xp|m−1≤|λ−at…t||xt|m−1+∑δti2…im=0δpi2…im=0|ati2…im||xt|m−1=(|λ−at…t|+rt(A)−|atp…p|)|xt|m−1. (3)

Multiplying (2) with (3), we yield

(2|apt…t|−rp(A))|atp…p||xt|m−1|xp|m−1≤|λ−ap…p|(|λ−at…t|+rt(A)−|atp…p|)|xt|m−1|xp|m−1

and

(2|apt…t|−rp(A))|atp…p|≤|λ−ap…p|(|λ−at…t|+rt(A)−|atp…p|). (4)

Hence, λ ∉ 𝓤_t,p(𝓐) and λ∈⋃i∈S,j∈S¯Ψi,j(A)).
If |x_t||x_p| ≠ 0 and |x_p| ≥ |x_t|, then |x_p| = max {|x_i| : i ∈ N} ≠ 0. Similar to the proof of Case 1, by (2.1) it holds that

(2|atp…p|−rt(A))|xp|m−1≤|λ−at…t||xt|m−1. (5)

Similarly, by the p-th equation of (2.1), we obtain

|apt…t||xt|m−1≤|λ−ap…p||xp|m−1+∑δpi2…im=0δti2…im=0|api2…im||xp|m−1=(|λ−ap…p|+rp(A)−|apt…t|)|xp|m−1. (6)

Multiplying (5) with (6), we yield

(2|atp…p|−rt(A))|apt…t|≤|λ−at…t|(|λ−ap…p|+rp(A)−|apt…t|). (7)

Hence, λ ∉ 𝓤_p,t(𝓐) and λ∈⋃i∈S¯,j∈SΨi,j(A).
If |x_t||x_p| = 0, |x_t| ≠ 0 and |x_p| = 0, then |x_t| ≥ |x_p|. From (2), we have 2|a_pt…t| − r_p(𝓐) ≤ 0. Since |λ − a_p…p|(|λ − a_t…t| + r_t(𝓐) − |a_tp…p|) ≥ 0, it holds that

(2|apt…t|−rp(A))|atp…p|≤|λ−ap…p|(|λ−at…t|+rt(A)−|atp…p|),

which implies λ ∉ 𝓤_t,p(𝓐) and λ∈⋃i∈S,j∈S¯Ψi,j(A)).

If |x_t||x_p| = 0, |x_p| ≠ 0 and |x_t| = 0, then |x_p| ≥ |x_t|. By (5), it holds that 2|a_tp…p| − r_t(𝓐) ≤ 0. Since |λ − a_t…t|(|λ − a_p…p| + r_p(𝓐) − |a_pt…t|) ≥ 0, we obtain

(2|atp…p|−rt(A))|apt…t|≤|λ−at…t|(|λ−ap…p|+rp(A)−|apt…t|),

which implies λ ∉ 𝓤_p,t(𝓐) and λ∈⋃i∈S¯,j∈SΨi,j(A).

Combining the discussions for the above three cases, we obtain λ∈⋃i∈S,j∈S¯Ψi,j(A))⋃⋃i∈S¯,j∈SΨi,j(A).

Next, we show the exclusion set 𝓤_i,j(𝓐) ⊆ 𝓚_i,j(𝓐). For any λ̃ ∈ 𝓤_t,p(𝓐), it holds that

(|λ~−at…t|+rt(A)−|atp…p|)|λ~−ap…p|<(2|apt…t|−rp(A))|atp…p|. (8)

The following argument is divided into two cases.

If (2|a_pt…t| − r_p(𝓐))|a_tp…p| ≤ 0, then 𝓤_t,p(𝓐) = ∅. Obviously, 𝓤_t,p(𝓐) ⊆ 𝓚_t,p(𝓐).
If (2|a_pt…t| − r_p(𝓐))|a_tp…p| > 0, from (8), we have

|λ~−at…t|+rt(A)−|atp…p||atp…p|⋅|λ~−ap…p|2|apt…t|−rp(A)<1. (9)

Noting that |a_pt…t| ≤ r_p(𝓐), i.e., 2|a_pt…t| − r_p(𝓐) ≤ r_p(𝓐), we obtain

|λ~−ap…p|rp(A)≤|λ~−ap…p|2|apt…t|−rp(A). (10)

Meanwhile,

|λ~−at…t|−(rt(A)−|atp…p|)|atp…p|≤|λ~−at…t|+rt(A)−|atp…p||atp…p|. (11)

It follows from (9), (10) and (11) that

|λ~−at…t|−(rt(A)−|atp…p|)|atp…p|⋅|λ~−ap…p|rp(A)≤1,

which implies

(|λ~−at…t|−(rt(A)−|atp…p|))|λ~−ap…p|≤|atp…p|rp(A).

So, λ̃ ∈ 𝓚_t,p(𝓐) and 𝓤_t,p(𝓐) ⊆ 𝓚_t,p(𝓐). For the arbitrariness of t, p, we have

λ∈⋃i∈S,j∈S¯Ki,j(A)∖Ui,j(A)=⋃i∈S,j∈S¯Ψi,j(A).

In a similar way, we also get λ∈⋃i∈S¯,j∈SKi,j(A)∖Ui,j(A)=⋃i∈S¯,j∈SΨi,j(A). So,

λ∈(⋃i∈S,j∈S¯Ψi,j(A))⋃(⋃i∈S¯,j∈SΨi,j(A))=ΨS(A)⊆KS(A).

□

The determinant of a tensor 𝓐, denoted by det(𝓐), is the resultant of the ordered system of homogeneous equations 𝓐x^m−1 = 0 of [4], and is closely related to the eigenvalue inclusion sets of a tensor 𝓐. From Theorem 2.2, by verifying 2|S|(n − |S|) sets 𝓚_i,j(𝓐) or 2|S|(n − |S|) sets 𝓤_i,j(𝓐), we obtain the following condition such that det(𝓐) ≠ 0. Compared with Corollary 3 of [24], Corollary 2.3 reduces computations and cuts down the validation conditions with det(𝓐) ≠ 0.

Corollary 2.3

Let 𝓐 be a complex tensor of order m and dimension n ≥ 2 and S be a nonempty proper subset of N. For i ∈ S, j ∈ S̄ and i ∈ S̄, j ∈ S, if either

(|ai…i|−(ri(A)−|aij…j|))|aj…j|>|aij…j|rj(A)

(|ai…i|+ri(A)−|aij…j|)|aj…j|<|aij…j|(2|aji…i|−rj(A)),

then det(𝓐) ≠ 0.

To obtain tighter inclusion sets than 𝓚^S(𝓐), Li et al. [33] established new S-type inclusion set for tensors. According to Section 2 of [33], for simplicity, we denote

ΔN={(i2,i3,…,im):each ij∈N for j=2,…,m},ΔS={(i2,i3,…,im):each ij∈S for j=2,…,m},

where S is nonempty proper subset of N, and then Δ^S = Δ^N ∖ Δ^S.

Given a tensor 𝓐 = (a_{i₁…i_m}), let

riΔS(A)=∑(i2,…,im)∈ΔSδii2…im=0|aii2…im|, riΔS¯(A)=∑(i2,…,im)∈ΔS¯|aii2…im|.

Obviously, ri(A)=riΔS(A)+riΔS¯(A).

Lemma 2.4

(Theorem 4 of [24]) Let 𝓐 be a complex tensor of order m and dimension n ≥ 2 and S be a nonempty proper subset of N. Then, all eigenvalues of 𝓐 are located in the union of the following sets:

σ(A)⊆ΩS(A)=(⋃i∈S,j∈S¯Ωi,jS(A))⋃(⋃i∈S¯,j∈SΩi,jS¯(A)),

where

Ωi,jS(A)={z∈C:|z−ai…i|(|z−aj…j|−rjΔS¯(A))≤ri(A)rjΔS(A)}

and

Ωi,jS¯(A)={z∈C:|z−ai…i|(|z−aj…j|−rjΔ S¯¯(A))≤ri(A)rjΔS¯(A)}.

Based on sharp S-type eigenvalue inclusion sets of [24], we are at the position to establish the following theorem.

Theorem 2.5

Let 𝓐 be a complex tensor of order m and dimension n ≥ 2 and S be a nonempty proper subset of N. Then, all eigenvalues of 𝓐 are located in the union of the following sets:

σ(A)⊆ΦS(A)=(⋃i∈S,j∈S¯Φi,jS(A))⋃(⋃i∈S¯,j∈SΦi,jS¯(A)),

where Φi,jS(A)=Ωi,jS(A)∖Vi,jS(A),Φi,jS¯(A)=Ωi,jS¯(A)∖Vi,jS¯(A) and

Vi,jS(A)={z∈C:(|z−ai…i|+ri(A)−|aij…j|)(|z−aj…j|+rjΔS¯(A))<|aij…j|(2|aji…i|−rjΔS(A))},Vi,jS¯(A)={z∈C:(|z−ai…i|+ri(A)−|aij…j|)(|z−aj…j|+rjΔS¯¯(A))<|aij…j|(2|aji…i|−rjΔS¯(A))}.

Furthermore, Φ^S(𝓐) ⊆ Ω^S(𝓐).

Proof

Let (λ, x) be an eigenvalue-eigenvector of 𝓐. Setting |x_t| = max {|x_i| : i ∈ S} and |x_p| = max {|x_i| : i ∈ S̄}, we obtain at least one of |x_t| and |x_p| is nonzero. The following argument is divided into three cases.

If |x_t||x_p| ≠ 0 and |x_t| ≥ |x_p|, then |x_t| = max {|x_i| : i ∈ N} ≠ 0. By the p-th equation of (1), we have

(λ−ap…p)xpm−1=∑(i2,…,im)∈ΔS¯δpi2…im=0api2…imxi2…xim+∑(i2,…,im)∈ΔSapi2…imxi2…xim=∑(i2,…,im)∈ΔS¯δpi2…im=0api2…imxi2…xim+apt…txtm−1+∑(i2,…,im)∈ΔSδti2…im=0api2…imxi2…xim

and

apt…txtm−1=(λ−ap…p)xpm−1−∑(i2,…,im)∈ΔS¯δpi2…im=0api2…imxi2…xim−∑(i2,…,im)∈ΔSδti2…im=0api2…imxi2…xim.

Taking modulus in the above equation and using the triangle inequality gives

|apt…t||xt|m−1≤|λ−ap…p||xp|m−1+∑(i2,…,im)∈ΔS¯δpi2…im=0|api2…im||xp|m−1+∑(i2,…,im)∈ΔSδti2…im=0|api2…im||xt|m−1=|λ−ap…p||xp|m−1+rpΔS¯(A)|xp|m−1+(rpΔS(A)−|apt…t|)|xt|m−1,

equivalently,

(2|apt…t|−rpΔS(A))|xt|m−1≤(|λ−ap…p|+rpΔS¯(A))|xp|m−1. (12)

Noting the t-th equation of (2.1), we obtain

(λ−at…t)xtm−1=∑δti2…im=0δpi2…im=0ati2…imxi2…xim+atp…pxpm−1

and

|atp…p||xp|m−1≤|λ−at…t||xt|m−1+∑δti2…im=0δpi2…im=0|ati2…im||xt|m−1,

equivalently,

|atp…p||xp|m−1≤(|λ−at…t|+rt(A)−|atp…p|)|xt|m−1. (13)

Multiplying (12) with (13), one has

|atp…p|(2|apt…t|−rpΔS(A))≤(|λ−at…t|+rt(A)−|atp…p|)(|λ−ap…p|+rpΔS¯(A)), (14)

which implies λ∉Vt,pS(A) and λ∈⋃i∈S,j∈S¯Ωi,jS(A).
If |x_t||x_p| ≠ 0 and |x_p| ≥ |x_t|, then |x_p| = max {|x_i| : i ∈ N} ≠ 0. Similar to the proof of Case 1, by the t-th equation of (2.1), we have

atp…pxpm−1=(λ−at…t)xtm−1−∑(i2,…,im)∈ΔS¯¯δti2…im=0ati2…imxi2…xim−∑(i2,…,im)∈ΔS¯δpi2…im=0ati2…imxi2…xim.

and

(2|atp…p|−rtΔS¯(A))|xp|m−1≤(|λ−at…t|+rtΔS¯¯(A))|xt|m−1. (15)

Recalling the p-th equation of (2.1), we obtain

|apt…t||xt|m−1≤|λ−ap…p||xp|m−1+∑δpi2…im=0δti2…im=0|api2…im||xp|m−1≤(|λ−ap…p|+rp(A)−|apt…t|)|xp|m−1. (16)

Multiplying (15) with (16) yields

|apt…t|(2|atp…p|−rtΔS¯(A))≤(|λ−ap…p|+rp(A)−|apt…t|)(|λ−at…t|+rtΔS¯¯(A)), (17)

which shows λ∉Vp,tS¯(A) and λ∈⋃i∈S¯,j∈SΦi,jS¯(A).
If |x_t||x_p| = 0, |x_t| ≠ 0 and |x_p| = 0, then |x_t| ≥ |x_p|. From (12), we obtain 2|apt…t|−rpΔS(A)≤0. Noting that (|λ−at…t|+rt(A)−|atp…p|)(|λ−ap…p|+rpΔS¯(A))≥0, we have

|atp…p|(2|apt…t|−rpΔS(A))≤(|λ−at…t|+rt(A)−|atp…p|)(|λ−ap…p|+rpΔS¯(A)),

which shows λ∉Vt,pS(A) and λ∈⋃i∈S,j∈S¯Ωi,jS(A).

If |x_t||x_p| = 0, |x_t| = 0 and |x_p| ≠ 0, then |x_p| ≥ |x_t|. Using (15), it holds that 2|atp…p|−rtΔS¯(A)≤0. From (|λ−ap…p|+rp(A)−|apt…t|)(|λ−at…t|+rtΔS¯¯(A))≥0, we deduce

|apt…t|(2|atp…p|−rtΔS¯(A))≤(|λ−ap…p|+rp(A)−|apt…t|)(|λ−at…t|+rtΔS¯¯(A)),

which implies λ∉Vp,tS¯(A) and λ∈⋃i∈S¯,j∈SΦi,jS¯(A).

To sum up above three discussions, we yield that λ∉Vt,pS(A)⋃Vp,tS¯(A).

Next, we establish Vt,pS(A)⊆Ωt,pS(A). For any λ~∈Vt,pS(A), we have

The following argument is divided into two cases.

If |atp…p|(2|apt…t|−rpΔS(A))≤0, then Vt,pS(A)=∅. Obviously, Vt,pS(A)⊆Ωt,pS(A).
If |atp…p|(2|apt…t|−rpΔS(A))>0, dividing (18) through by |atp…p|(2|apt…t|−rpΔS(A)), we have

|λ~−at…t|+rt(A)−|atp…p||atp…p|⋅|λ~−ap…p|+rpΔS¯(A)2|apt…t|−rpΔS(A)<1. (19)

From |apt…t|≤rpΔS(A), we know 2|apt…t|−rpΔS(A)≤rpΔS(A). Hence,

|λ~−ap…p|−rpΔS¯(A)rpΔS(A)≤|λ~−ap…p|+rpΔS¯(A)2|apt…t|−rpΔS(A). (20)

By |a_tp…p| ≤ r_t(𝓐), it holds that

|λ~−at…t|rt(A)≤|λ~−at…t|+rt(A)−|atp…p||atp…p|. (21)

It follows from (19), (20) and (21) that

|λ~−at…t|rt(A)⋅|λ~−ap…p|−rpΔS¯(A)rpΔS(A)≤1,

which implies

|λ~−at…t|(|λ~−ap…p|−rpΔS¯)≤rt(A)rpΔS(A)

and Vt,pS(A)⊆Ωt,pS(A). For the arbitrariness of t, p, we have

λ∈⋃i∈S,j∈S¯Ωi,jS(A)∖Vi,jS(A)=⋃i∈S,j∈S¯Φi,jS(A).

Similarly, we can obtain

λ∈⋃i∈S¯,j∈SΩi,jS¯(A)∖Vi,jS¯(A)=⋃i∈S¯,j∈SΦi,jS¯(A).

Furthermore,

λ∈(⋃i∈S,j∈S¯Φi,jS(A))⋃(⋃i∈S¯,j∈SΦi,jS¯(A))=ΦS(A)⊆ΩS(A).

□

Corollary 2.6

Let 𝓐 be a complex tensor of order m and dimension n ≥ 2 and S be a nonempty proper subset of N. For i ∈ S, j ∈ S̄ and i ∈ S̄, j ∈ S, if either

|ai…i|(|aj…j|−rjΔS¯(A))>ri(A)rjΔS(A)

(|ai…i|+ri(A)−|aij…j|)(|aj…j|+rjΔS¯(A))<|aij…j|(2|aji…i|−rjΔS(A)),

then det(𝓐) ≠ 0.

3 Numerical examples

The following example exhibits the superiority of the results given in Theorem 2.2.

Example 3.1

Consider 3 order 3 dimensional tensor 𝓐 = (a_ijk) defined by

aijk=a222=a133=2;a333=32;a122=a211=a311=1;a233=12;aijk=0, otherwise.

For simplicity, we take λ as a real number, where λ is an eigenvalue of 𝓐.

According to Theorem 2 of [24], we have

Ω(A)=⋃i∈NΩi(A)={λ∈C:−3≤λ≤3.5}.

From Theorem 4 of [24], we obtain

Θ(A)=⋃i,j∈N,i≠jΘi,j(A)={λ∈C:−2.345≤λ≤3.386},

where

Θ1,2∪Θ1,3={λ∈C:−2.345≤λ≤−0.186}∪{λ∈C:2.137≤λ≤3.225}Θ2,1∪Θ2,3={λ∈C:−1.137≤λ≤3.386}Θ3,1∪Θ3,2={λ∈C:−1.137≤λ≤−0.5}∪{λ∈C:2≤λ≤2.5}K1,2={λ∈C:−2.345≤λ≤3.225},K1,3={λ∈C:−1.637≤λ≤2.686}K2,1={λ∈C:−1.137≤λ≤3.386},K2,3={λ∈C:0.5≤λ≤3.281}K3,1={λ∈C:−1.137≤λ≤2.637},K3,2={λ∈C:0.5≤λ≤2.5}U1={λ∈C:−0.186<λ<2.137},U3={λ∈C:−0.5<λ<2},U2=∅.

Without loss of generality, we select S = {3}, S̄ = {1, 2}. According to Lemma 2.1 and Theorem 2.2, we get

KS(A)=(K3,1(A)∪K3,2(A))⋃(K1,3(A)∪K2,3(A))={λ∈C:−1.637≤λ≤3.281}

and

ΨS(A)=(Ψ3,1(A)∪Ψ3,2(A))⋃(Ψ1,3(A)∪Ψ2,3(A))={λ∈C:−1.637≤λ≤−0.186}∪{λ∈C:0.5≤λ≤3.281}.

From the fact that

Ω(A)={λ∈C:−3≤λ≤3.5},Θ(A)={λ∈C:−2.345≤λ≤3.386}

and

ΨS(A)={λ∈C:−1.637≤λ≤−0.186}∪{λ∈C:0.5≤λ≤3.281},

we conclude that the result given in Theorem 2.2 is more accurate than Theorem 2 and Theorem 4 of [24].

The following example exhibits the efficiency of the new inclusion sets given in Theorem 2.5.

Example 3.2

Consider 3 order 3 dimensional tensor 𝓐 = (a_ijk) defined by

aijk=a111=12;a222=14;a333=−3;a122=2;a112=a131=110;a211=a232=18;a311=a332=116;aijk=0, otherwise.

For simplicity, we take λ as a real number, where λ is an eigenvalue of 𝓐.

According to Theorem 2 of [24], we have

Ω(A)=⋃i∈NΩi(A)={λ∈C:−3.125≤λ≤−2.875}⋃{λ∈C:−1.7≤λ≤2.7}.

According to Theorem 4 of [24], we have

Θ(A)=⋃i,j∈N,i≠jΘi,j(A)={λ∈C:−3.125≤λ≤−2.875}⋃{λ∈C:−1.7≤λ≤2.7}.

Without loss of generality, we choose S = {1}, S̄ = {2, 3}. According to Lemma 2.4 and Theorem 2.5, we obtian

ΩS(A)=(Ω1,2S(A)∪Ω1,3S(A))⋃(Ω2,1S¯(A)∪Ω3,1S¯(A))={λ∈C:−3.101≤λ≤−2.897}⋃{λ∈C:−0.433≤λ≤1.217}

and

ΦS(A)=(Φ1,2S(A)∪Φ1,3S(A))⋃(Φ2,1S¯(A)∪Φ3,1S¯(A))={λ∈C:−3.101≤λ≤−2.897}⋃{λ∈C:−0.433≤λ≤0.012}⋃{λ∈C:0.222≤λ≤1.217}.

So, we conclude that the result given in Theorem 2.5 is much sharper than Theorem 2 and Theorem 4 of [24] in some cases.

Remark 3.3

It is worth noting that Ω^S(𝓐) ⊆ 𝓚^S(𝓐) ⊆ 𝓚(𝓐) ⊆ Γ(𝓐) by Theorem 2.3 of [23] and Theorem 6 of [24]. However, Φ^S(𝓐) ⊆ Ψ^S(𝓐) may not hold in general because the exclusion sets Vi,jS(A)⊆Ui,j(A).

In order to compare two new S-type inclusion sets, we further compute the above two examples. We apply Theorem 2.2 to example 3.2 and obtain that

ΨS(A)={λ∈C:−3.125≤λ≤−2.875}∪{λ∈C:−1.7≤λ≤2.7},

which shows that the result of Theorem 2.5 is sharper than that of Theorem 2.2 in this case.

On the other hand, we apply Theorem 2.5 to example 3.1 and obtain that

ΦS(A)={λ∈C:−1.345≤λ≤3.386},

which shows that the result of Theorem 2.2 is tighter than that of Theorem 2.5 in this case. Thus, Theorem 2.2 and Theorem 2.5 have their own advantages.

4 Conclusions

In the paper, we focused on S-type eigenvalue exclusion sets for tensors. S-type eigenvalue exclusion sets have two advantages: (i) all eigenvalues of tensor are not contained; (ii) they may reduce computations and achieve more accurate numerical results. Based on characterizations of S-type eigenvalue exclusion sets, we proposed new S-type inclusion sets Ψ^S(𝓐) and Φ^S(𝓐). Furthermore, we showed that new S-type eigenvalue inclusion sets are sharper than those of results in [23, 24, 34] via two examples. It is worth noting that the proper selection of S has a great influence on the numerical effect of S-type inclusion sets. Therefore, how to select the proper S is our further research.

Acknowledgement

This work is supported by the Natural Science Foundation of China (11671228) and the Natural Science Foundation of Shandong Province (ZR2016AM10).

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Received: 2018-10-19

Accepted: 2019-08-26

Published Online: 2019-10-13

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0090

Keywords for this article

S-type eigenvalue inclusion sets; S-type eigenvalue exclusion sets; Brauer-type eigenvalue inclusion sets

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