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Geršhgorin-type theorems for Z1-eigenvalues of tensors with applications

  • Xiaowei Shen , Qian Hao , Jun He EMAIL logo and Guangbin Wang
Published/Copyright: April 1, 2025
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In this article, we present several Geršhgorin-type theorems for Z 1 -eigenvalues of tensors, which improve the results provided by Wang et al. (Some upper bounds on Z t -eigenvalues of tensors, Appl. Math. Comput. 329 (2018), 266–277). Bounds for the largest Z 1 -eigenvalue of nonnegative tensors are given. Furthermore, based on the Geršhgorin-type theorems, we introduce some sufficient criteria for the positivity of even order tensors.

Keywords: H-eigenvalue; Geršhgorin theorem; Z 1-eigenvalue; Z 2-eigenvalue; bounds
MSC 2010: 15A69; 15A18; 65F10; 65F15

1 Introduction

K = ( k j 1 j 2 … j m ) ∈ T [ m , ι ] is called a real square tensor if j 1 , j 2 , … , j m ∈ [ ι ] = { 1 , 2 , … , ι } , if

k j 1 j 2 … j m = k j ϕ ( 1 ) j ϕ ( 2 ) … j ϕ ( m ) , ϕ ∈ ϕ m ,

where ϕ m is the permutation group of m indices, K is called symmetric, denoted by K ∈ ST [ m , ι ] [1]. If

K μ m − 1 = ∑ j 2 , … , j m = 1 n k j j 2 … j m μ j 2 … μ j m j ∈ [ ι ] , K μ m − 1 = τ μ [ m − 1 ] ,

where τ ∈ R and μ ∈ R n , μ [ m − 1 ] ≔ ( μ 1 m − 1 , μ 2 m − 1 , … , μ ι m − 1 ) ⊤ , then ( τ , μ ) is called an H -eigenpair of K . If

K μ m − 1 = τ μ , ‖ μ ‖ 2 = ∑ j ∈ [ ι ] μ j 2 = 1 ,

then ( τ , μ ) is called a Z 2 -eigenpair of K . If

K μ m − 1 = τ μ , ‖ μ ‖ 1 = ∑ j ∈ [ ι ] ∣ μ j ∣ = 1 ,

then ( τ , μ ) is called a Z 1 -eigenpair of K [1–3].

Eigenvalues of tensors play a key role in practical applications, such as best rank-one approximations [4], transition probability tensors in Markov chains [3], automatic control [5–10], spectral hypergraph theory [11–14], etc. Kolda and Mayo introduced the power method to compute tensor eigenpairs [15], and the so-called homotopy methods are proposed in [16]. Many eigenvalue inclusion theorems for tensors are explored by researchers, including the inclusion sets for H -eigenvalues [17–23], and the inclusion sets for Z 2 -eigenvalues [24–35]. When K is an even order tensor, sufficient criteria for the positivity of K are discussed based on these inclusion theorems for H -eigenvalues or Z 2 -eigenvalues.

The Z 1 -eigenpairs are useful for computing the limiting probability distribution in higher order Markov chain [3] and the PageRank vector in multilinear PageRank models [36], best rank-one approximations in statistical data analysis [4], etc. Some effective algorithms for finding Z 1 -eigenvalues and the corresponding eigenvectors have been proposed [4,15]. However, it is difficult to compute all Z 1 -eigenvalues, even the smallest Z 1 -eigenvalue when m and ι are very large [37]. Chang and Tan Zhang [3] also showed that ( τ , μ ) is a Z 1 -eigenpair of K if and only if ( τ ‖ μ ‖ 2 m − 2 , μ ‖ μ ‖ 2 ) is a Z 2 -eigenpair of K . Therefore, together with the results of Theorem 5 in [1], the following results can be obtained directly.

Theorem 1

Let K ∈ ST [ m , ι ] be an even order tensor. Then, K is a positive definite tensor if and only if all the Z 1 -eigenvalues of K are positive.

There are just three papers concerning about the inclusion theorems for Z 1 -eigenvalues of tensors [26,38,39]. Unfortunately, when K is an even order tensor, no sufficient criteria for the positivity of K are discussed based on the inclusion theorems for Z 1 -eigenvalues.

In summary, the main contributions of this article are as follows:

1. We further investigate the properties of Z 1 -eigenpair, and several new and refined inclusion intervals for Z 1 -eigenvalue of tensors are given. If K is nonnegative, denoted by K ∈ T + [ m , ι ] , bounds for the largest Z 1 -eigenvalue of K are also discussed.

2. Sufficient criteria for the positivity of tensors are first investigated based on the inclusion intervals for Z 1 -eigenvalue of tensors, and these criteria perform very well in some cases.

In Section 2, we obtain the Geršhgorin-type theorems for Z 1 -eigenvalues of tensors, including the Geršhgorin theorem, Brauer theorem, and S-type theorem. The relationships between these sets are also presented. If K ∈ T + [ m , ι ] , bounds for the largest Z 1 -eigenvalue of K are given in Section 3. Sufficient criteria for the positivity of tensors are discussed in Section 4.

2 Geršhgorin-type theorems for Z 1 -eigenvalues of tensors

Assume that ϕ ( j 2 , … , j m ) denotes the set of all permutations of ( j 2 , … , j m ) and K ^ = ( k ^ j 1 j 2 … j m ) ∈ T [ m , ι ] with

(1) ∑ j 2 , … , j m ∈ [ ι ] k ^ j j 2 … j m = ∑ ( j 2 , … , j m ) ∈ ϕ ( j 2 , … , j m ) k j j 2 … j m .

Remark 1

Let ( τ , μ ) be a Z 1 -eigenpair of K , by the fact K μ m − 1 = K ^ μ m − 1 , we have σ Z 1 ( K ) = σ Z 1 ( K ^ ) , where σ Z 1 ( K ) is the set which contains all the Z 1 -eigenvalues of K .

Remark 2

In this remark, we give an example to explain why we use the operation in (1). Let K = ( k j 1 j 2 j 3 ) ∈ T [ 3,2 ] with

k 111 = 1 , k 112 = 21 , k 121 = − 20 , k 222 = 1 , k 212 = 100 , k 221 = − 101 ,

and other entries be all zero, K ^ = ( k ^ j 1 j 2 j 3 ) ∈ T [ 3,2 ] with

k ^ 111 = 1 , k ^ 112 = 1 , k ^ 121 = 0 , k ^ 222 = 1 , k ^ 212 = 0 , k ^ 221 = − 1 .

Obviously, ∣ k ^ j 1 j 2 j 3 ∣ ≤ ∣ k j 1 j 2 j 3 ∣ , which means that if we use the operation in (1) properly, all the bounds given in [38,39] are expected to be sharper.

We give our main results as follows.

Theorem 2

(Geršhgorin theorem for Z 1 -eigenvalues of tensors) Let K = ( k j 1 j 2 … j m ) ∈ T [ m , ι ] , K ¯ s = ( k ¯ j j s ) = ( max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ( ∣ k ^ j j 2 … j s … j m ∣ ) ) ∈ T [ 2 , ι ] , τ ∈ σ Z 1 ( K ) , then

∣ τ ∣ ≤ ω 1 ≔ min s ∈ [ m ] \ { 1 } ( max j ∈ [ ι ] R j ( K ¯ s ) ) ,

where

R j ( K ¯ s ) = ∑ j s ∈ [ ι ] k ¯ j j s .

Proof

Let ( τ , μ ) be a Z 1 -eigenpair of K and μ = ( μ 1 , … , μ ι ) ⊤ , that is,

(2) K μ m − 1 = τ μ , ‖ μ ‖ 1 = 1 .

Let

∣ μ p ∣ = max { ∣ μ j ∣ , j ∈ [ ι ] } .

Obviously, 1 ≥ ∣ μ p ∣ > 0 , we have

(3) τ μ p = ∑ j 2 , … , j m ∈ [ ι ] k p j 2 … j m μ j 2 … μ j m = ∑ j 2 , … , j m ∈ [ ι ] k ^ p j 2 … j m μ j 2 … μ j m .

Let s ∈ [ m ] \ { 1 } be an arbitrary index, taking modulus in (3), we can obtain

(4) ∣ τ ∣ ∣ μ p ∣ = ∑ j 2 , … , j m ∈ [ ι ] k p j 2 … j m μ j 2 … μ j m = ∑ j 2 , … , j m ∈ [ ι ] k ^ p j 2 … j m μ j 2 … μ j m ≤ ∑ j 2 , … , j m ∈ [ ι ] ∣ k ^ p j 2 … j m ∣ ∣ μ j 2 ∣ … ∣ μ j m − 1 ∣ ∣ μ j m ∣ = ∑ j m ∈ [ ι ] ∑ j 2 , … , j m − 1 ∈ [ ι ] ∣ k ^ p j 2 … j m − 1 j m ∣ ∣ μ j 2 ∣ … ∣ μ j m − 1 ∣ ∣ μ j m ∣ ≤ max j m ∈ [ ι ] ∑ j 2 , … , j m − 1 ∈ [ ι ] ∣ k ^ p j 2 … j m − 1 j m ∣ ∣ μ j 2 ∣ … ∣ μ j m − 1 ∣ ∑ j m ∈ [ ι ] ∣ μ j m ∣ = max j m ∈ [ ι ] ∑ j 2 , … , j m − 1 ∈ [ ι ] ∣ k ^ p j 2 … j m − 1 j m ∣ ∣ μ j 2 ∣ … ∣ μ j m − 1 ∣ ⋮ ≤ max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] ∣ k ^ p j 2 … j s … j m ∣ ∣ μ j s ∣ ≤ max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] ∣ k ^ p j 2 … j s … j m ∣ ∣ μ p ∣ .

Therefore,

∣ τ ∣ ≤ max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] ∣ k ^ p j 2 … j s … j m ∣ = R p ( K ¯ s ) .

Since s ∈ [ m ] \ { 1 } is an arbitrary index, we obtain

□ ∣ τ ∣ ≤ min s ∈ [ m ] \ { 1 } ( max j ∈ [ ι ] R j ( K ¯ s ) ) .

Remark 3

If s = { 1 } , we can obtain

(5) ∣ τ ∣ = ∣ τ ∣ ∑ j 1 ∈ [ ι ] ∣ μ j 1 ∣ = ∑ j 1 , … , j m ∈ [ ι ] k ^ j 1 j 2 … j m μ j 2 … μ j m ≤ ∑ j 1 , … , j m ∈ [ ι ] ∣ k ^ j 1 j 2 … j m − 1 j m ∣ ∣ μ j 2 ∣ … ∣ μ j m − 1 ∣ ∣ μ j m ∣ ⋮ ≤ max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j 1 , j s ∈ [ ι ] ∣ k ^ j 1 j 2 … j s … j m ∣ ∣ μ j s ∣ ≤ max j 2 , … , j m ∈ [ ι ] ∑ j 1 ∈ [ ι ] ∣ k ^ j 1 j 2 … j m ∣ = R 1 ( K ^ ) ,

together with the result in Theorem 2, we have

(6) ∣ τ ∣ ≤ min { R 1 ( K ^ ) , ω 1 } .

Let k ^ j j 2 j 3 = k j j 2 j 3 , the result in (6) coincides with the result in Theorem 2.1 in [38]. In the following theorems, we improve the result obtained in Theorem 2, unfortunately, the result in (5) cannot be improved, so we will not discuss it in the following study.

Remark 4

Considering the example in Remark 2, by Theorem 2.1 in [38], we have

C 1 ( K ) = max j 2 , j 3 ∈ [ ι ] ∑ j 1 ∈ [ ι ] ∣ k j 1 j 2 j 3 ∣ = ∣ k 112 ∣ + ∣ k 212 ∣ = 121 , C 2 ( K ) = max j 1 , j 3 ∈ [ ι ] ∑ j 2 ∈ [ ι ] ∣ k j 1 j 2 j 3 ∣ = ∣ k 221 ∣ = 101 , C 3 ( K ) = max j 1 , j 2 ∈ [ ι ] ∑ j 3 ∈ [ ι ] ∣ k j 1 j 2 j 3 ∣ = ∣ k 221 ∣ + ∣ k 222 ∣ = 102 ,

then

∣ τ ∣ ≤ min { C 1 ( K ) , C 2 ( K ) , C 3 ( K ) } = 101 .

By (6), we have

R 1 ( K ) = max j 2 , j 3 ∈ [ ι ] ∑ j 1 ∈ [ ι ] ∣ k ^ j 1 j 2 j 3 ∣ = ∣ k ^ 111 ∣ = 1 , ω 1 ≔ min s ∈ [ m ] \ { 1 } ( max j ∈ [ ι ] R j ( K ¯ s ) ) = 1 ,

then

∣ τ ∣ ≤ min { R 1 ( K ^ ) , ω 1 } = 1 .

In fact, σ Z 1 ( K ) = σ Z 1 ( K ^ ) = { 1 } . From this example, we can see that if we use the operation in (1) properly, the result in Theorem 2 is sharper.

Theorem 3

(Brauer theorem for Z 1 -eigenvalues of tensors) Let K = ( k j 1 j 2 … j m ) ∈ T [ m , ι ] , τ ∈ σ Z 1 ( K ) , then

∣ τ ∣ ≤ ω 2 ≔ min s ∈ [ m ] \ { 1 } max j , t ∈ [ ι ] , j ≠ t 1 2 ( k ¯ j j + k ¯ t t + ( k ¯ j j − k ¯ t t ) 2 + 4 r j ( K ¯ s ) r t ( K ¯ s ) ) ,

where

r j ( K ¯ s ) = R j ( K ¯ s ) − k ¯ j j .

Proof

Let

∣ μ p ∣ = max { ∣ μ j ∣ , j ∈ [ ι ] } , ∣ μ q ∣ = max { ∣ μ j ∣ , j ∈ [ ι ] , j ≠ p } .

Obviously, 1 ≥ ∣ μ p ∣ ≥ ∣ μ q ∣ ≥ 0 , considering the p th equation in (2), we have

∣ τ ∣ ∣ μ p ∣ ≤ max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] ∣ k ^ p j 2 … j s … j m ∣ ∣ μ j s ∣ = max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ( ∣ k ^ p j 2 … p … j m ∣ ∣ μ p ∣ ) + max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] , j s ≠ p ∣ k ^ p j 2 … j s … j m ∣ ∣ μ j s ∣ ≤ max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ( ∣ k ^ p j 2 … p … j m ∣ ∣ μ p ∣ ) + max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] , j s ≠ p ∣ k ^ p j 2 … j s … j m ∣ ∣ μ q ∣ ,

which means

( ∣ τ ∣ − max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ( ∣ k ^ p j 2 … p … j m ∣ ) ) ∣ μ p ∣ ≤ max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] , j s ≠ p ∣ k ^ p j 2 … j s … j m ∣ ∣ μ q ∣ .

Therefore,

(7) ( ∣ τ ∣ − k ¯ p p ) ∣ μ p ∣ ≤ ∑ j s ∈ [ ι ] , j s ≠ p k ¯ p j s ∣ μ q ∣ = r p ( K ¯ s ) ∣ μ q ∣ .

Similarly, considering the q th equation in (2), we have

(8) ( ∣ τ ∣ − k ¯ q q ) ∣ μ q ∣ ≤ ∑ j s ∈ [ ι ] , j s ≠ q k ¯ q j s ∣ μ p ∣ = r q ( K ¯ s ) ∣ μ p ∣ .

Two cases are considered as follows.

Case I: If ∣ τ ∣ − k ¯ p p > 0 or ∣ τ ∣ − k ¯ q q > 0 .

(i) If ∣ μ q ∣ > 0 , multiplying inequalities (7) and (8),

( ∣ τ ∣ − k ¯ p p ) ( ∣ τ ∣ − k ¯ q q ) ≤ r p ( K ¯ s ) r q ( K ¯ s ) ,

which means

∣ τ ∣ ≤ 1 2 ( k ¯ p p + k ¯ q q + ( k ¯ p p − k ¯ q q ) 2 + 4 r p ( K ¯ s ) r q ( K ¯ s ) ) .

(ii) If ∣ μ q ∣ = 0 , we have

(9) ∣ τ ∣ ≤ k ¯ p p ≤ 1 2 ( k ¯ p p + k ¯ q q + ( k ¯ p p − k ¯ q q ) 2 + 4 r p ( K ¯ s ) r q ( K ¯ s ) ) .

Case II: If ∣ τ ∣ − k ¯ p p < 0 and ∣ τ ∣ − k ¯ q q < 0 , then

∣ τ ∣ < k ¯ q q ≤ 1 2 ( k ¯ p p + k ¯ q q + ( k ¯ p p − k ¯ q q ) 2 + 4 r p ( K ¯ s ) r q ( K ¯ s ) ) .

Together with (9), we have

∣ τ ∣ ≤ 1 2 ( k ¯ p p + k ¯ q q + ( k ¯ p p − k ¯ q q ) 2 + 4 r p ( K ¯ s ) r q ( K ¯ s ) ) .

From the arbitrariness of s , we obtain

□ ∣ τ ∣ ≤ min s ∈ [ m ] \ { 1 } max j , t ∈ [ ι ] , j ≠ t 1 2 ( k ¯ j j + k ¯ t t + ( k ¯ j j − k ¯ t t ) 2 + 4 r j ( K ¯ s ) r t ( K ¯ s ) ) .

Theorem 4

(S-type theorem for Z 1 -eigenvalues of tensors) Let S be a nonempty subset of [ ι ] , S ¯ = [ ι ] \ S , and ι ≥ 2 , K = ( k j 1 j 2 … j m ) ∈ T [ m , ι ] , τ ∈ σ Z 1 ( K ) , then

∣ τ ∣ ≤ ω 3 ≔ min s ∈ [ m ] \ { 1 } min S ⊆ [ ι ] max j ∈ S , t ∈ S ¯ 1 2 ( r j S ( K ¯ s ) + r t S ¯ ( K ¯ s ) + ( r j S ( K ¯ s ) − r t S ¯ ( K ¯ s ) ) 2 + 4 r j S ¯ ( K ¯ s ) r t S ( K ¯ s ) ) ,

where

r j S ( K ¯ s ) = ∑ j s ∈ S k ¯ j j s , r j S ¯ ( K ¯ s ) = ∑ j s ∈ S ¯ k ¯ j j s , r t S ( K ¯ s ) = ∑ j s ∈ S k ¯ t j s , r t S ¯ ( K ¯ s ) = ∑ j s ∈ S ¯ k ¯ t j s .

Proof

Let

∣ μ p ∣ = max { ∣ μ j ∣ , j ∈ S } , ∣ μ q ∣ = max { ∣ μ t ∣ , t ∈ S ¯ } .

Obviously, ∣ μ p ∣ ≥ 0 , ∣ μ q ∣ ≥ 0 , and at least one of ∣ μ p ∣ , ∣ μ q ∣ greater than 0. Considering the p th equation in (2), we obtain

(10) ( ∣ τ ∣ − r p S ( K ¯ s ) ) ∣ μ p ∣ ≤ r p S ¯ ( K ¯ s ) ∣ μ q ∣ .

Considering the q th equation in (2), we have

(11) ( ∣ τ ∣ − r q S ¯ ( K ¯ s ) ) ∣ μ q ∣ ≤ r q S ( K ¯ s ) ∣ μ p ∣ .

Two cases are considered as follows.

Case I: If ∣ μ p ∣ > 0 and ∣ μ q ∣ > 0 .

(i) If ∣ τ ∣ − r p S ( K ¯ s ) > 0 or ∣ τ ∣ − r q S ¯ ( K ¯ s ) > 0 , multiplying inequalities (10) and (11),

(12) ( ∣ τ ∣ − r p S ( K ¯ s ) ) ( ∣ τ ∣ − r q S ¯ ( K ¯ s ) ) ≤ r p S ¯ ( K ¯ s ) r q S ( K ¯ s ) ,

which means

∣ τ ∣ ≤ 1 2 ( r p S ( K ¯ s ) + r q S ¯ ( K ¯ s ) + ( r p S ( K ¯ s ) − r q S ¯ ( K ¯ s ) ) 2 + 4 r p S ¯ ( K ¯ s ) r q S ( K ¯ s ) ) .

(ii) If ∣ τ ∣ − r p S ( K ¯ s ) < 0 and ∣ τ ∣ − r q S ¯ ( K ¯ s ) < 0 , then

∣ τ ∣ < r p S ( K ¯ s ) ≤ 1 2 ( r p S ( K ¯ s ) + r q S ¯ ( K ¯ s ) + ( r p S ( K ¯ s ) − r q S ¯ ( K ¯ s ) ) 2 + 4 r p S ¯ ( K ¯ s ) r q S ( K ¯ s ) ) , ∣ τ ∣ < r q S ¯ ( K ¯ s ) ≤ 1 2 ( r p S ( K ¯ s ) + r q S ¯ ( K ¯ s ) + ( r p S ( K ¯ s ) − r q S ¯ ( K ¯ s ) ) 2 + 4 r p S ¯ ( K ¯ s ) r q S ( K ¯ s ) ) .

Case II: If one of ∣ μ p ∣ , ∣ μ q ∣ equal to 0, assume that ∣ μ p ∣ > 0 and ∣ μ q ∣ = 0 . Then, ∣ τ ∣ − k ¯ p p < 0 , which means

∣ τ ∣ < r p S ( K ¯ s ) ≤ 1 2 ( r p S ( K ¯ s ) + r q S ¯ ( K ¯ s ) + ( r p S ( K ¯ s ) − r q S ¯ ( K ¯ s ) ) 2 + 4 r p S ¯ ( K ¯ s ) r q S ( K ¯ s ) ) .

From the arbitrariness of s and S , we obtain

□ ∣ τ ∣ ≤ min s ∈ [ m ] \ { 1 } min S ⊆ [ ι ] max j ∈ S , t ∈ S ¯ 1 2 ( r j S ( K ¯ s ) + r t S ¯ ( K ¯ s ) + ( r j S ( K ¯ s ) − r t S ¯ ( K ¯ s ) ) 2 + 4 r j S ¯ ( K ¯ s ) r t S ( K ¯ s ) ) .

Now, we discuss the relationships between ω 1 , ω 2 , and ω 3 .

Theorem 5

Let K = ( k j 1 j 2 … j m ) ∈ T [ m , ι ] , then

ω 3 ≤ ω 2 ≤ ω 1 .

Proof

For ∀ j , t ∈ [ ι ] , t ≠ j , assume that

k ¯ j j + r j ( K ¯ s ) ≥ k ¯ t t + r t ( K ¯ s ) ,

then

k ¯ j j + r j ( K ¯ s ) − k ¯ t t ≥ r t ( K ¯ s ) ≥ 0 ,

and

1 2 ( k ¯ j j + k ¯ t t + ( k ¯ j j − k ¯ t t ) 2 + 4 r j ( K ¯ s ) r t ( K ¯ s ) ) ≤ 1 2 ( k ¯ j j + k ¯ t t + ( k ¯ j j − k ¯ t t ) 2 + 4 r j ( K ¯ s ) ( k ¯ j j + r j ( K ¯ s ) − k ¯ t t ) ) = k ¯ j j + r j ( K ¯ s ) = R j ( K ¯ s ) .

Therefore,

ω 2 = min s ∈ [ m ] \ { 1 } max j , t ∈ [ ι ] , j ≠ t 1 2 ( k ¯ j j + k ¯ t t + ( k ¯ j j − k ¯ t t ) 2 + 4 r j ( K ¯ s ) r t ( K ¯ s ) ) ≥ ω 1 = min s ∈ [ m ] \ { 1 } ( max j ∈ [ ι ] R j ( K ¯ s ) ) .

Assume that ∣ μ p ∣ > ∣ μ q ∣ , S = { j } , from (10), we have

(13) ∣ τ ∣ − k ¯ j j ≤ r j ( K ¯ s ) .

And from (12), we have

( ∣ τ ∣ − k ¯ j j ) ( ∣ τ ∣ − k ¯ t t − r t ( K ¯ s ) + k ¯ t j ) ≤ r j ( K ¯ s ) k ¯ t j ,

together with (13), we can obtain

( ∣ τ ∣ − k ¯ j j ) ( ∣ τ ∣ − k ¯ t t ) ≤ r j ( K ¯ s ) k ¯ t j + ( ∣ τ ∣ − k ¯ j j ) ( r t ( K ¯ s ) − k ¯ t j ) ≤ r j ( K ¯ s ) k ¯ t j + r j ( K ¯ s ) ( r t ( K ¯ s ) − k ¯ t j ) = r j ( K ¯ s ) r t ( K ¯ s ) ,

which means that if ∣ τ ∣ ≤ ω 3 , then ∣ τ ∣ ≤ ω 2 . Therefore, ω 3 ≤ ω 2 .□

Remark 5

Let K = K ^ , by (5) and Theorems 2, 3, and 4, we can obtain

(14) ∣ τ ∣ ≤ min { R 1 ( K ^ ) , ω 3 } ≤ min { R 1 ( K ^ ) , ω 2 } ≤ min { R 1 ( K ^ ) , ω 1 } ,

which means that, the upper bound min { R 1 ( K ^ ) , ω 3 } in (14) is always better than the bounds in [38,39].

Example 1

Let K = ( k j 1 j 2 j 3 ) ∈ T [ 3,3 ] with

k 111 = 1 , k 112 = 11 , k 121 = − 10 , k 222 = 2 , k 212 = 50 , k 221 = − 51 , k 333 = 3 , k 323 = − 100 , k 332 = 102 ,

and other entries be all zero. By computations, we have σ Z 1 ( K ) = σ Z 1 ( K ^ ) = { 0.75 } . We compute the bounds for ∣ τ ∣ in Table 1.

(15) k ^ j j 2 j 3 = 1 2 ∑ ( j 2 , j 3 ) ∈ ϕ ( j 2 , j 3 ) k j j 2 j 3 ,

then, K ^ = ( k ^ j 1 j 2 j 3 ) ∈ T [ 3,3 ] with

k ^ 111 = 1 , k ^ 112 = 0.5 , k ^ 121 = 0.5 , k ^ 222 = 2 , k ^ 212 = − 0.5 , k ^ 221 = − 0.5 , k ^ 333 = 3 , k ^ 323 = 1 , k ^ 332 = 1 ,

and other entries be all zero. From Table 1, it can be found that the upper bound given in Theorem 4 is best than other bounds given in the literature.

Table 1

Different bounds for ∣ τ ∣

Theorem 2.1 of [26] ∣ τ ∣ ≤ 163
Theorem 2.5 of [39] ∣ τ ∣ ≤ 102
Theorem 2.1 of [38] ∣ τ ∣ ≤ 102
Theorem 2 ∣ τ ∣ ≤ 4
Theorem 3 ∣ τ ∣ ≤ 3.366
Theorem 4, S = { 3 } ∣ τ ∣ ≤ 3

Example 2

Let K = ( k j 1 j 2 j 3 ) ∈ T [ 3,3 ] which are randomly generated 1,000 tensors with the elements of each tensor are from − 10 to 10. Let

k ^ j j 2 j 3 = 1 2 ∑ ( j 2 , j 3 ) ∈ ϕ ( j 2 , j 3 ) k j j 2 j 3 ,

Different bounds for ∣ τ ∣ are shown in Figures 1 and 2. From Figures 1 and 2, we can see that bounds obtained in Theorem 4 are always better than the bounds given in [38,39].

Figure 1 Different upper bounds for ∣ τ ∣ | \tau | in Example 2.
Figure 1

Different upper bounds for ∣ τ ∣ in Example 2.

Figure 2 Boxplot for different upper bounds for ∣ τ ∣ | \tau | in Example 2.
Figure 2

Boxplot for different upper bounds for ∣ τ ∣ in Example 2.

3 Bounds for the largest Z 1 -eigenvalue of nonnegative tensors

The definitions of weakly symmetric and irreducible are introduced in [33]. If K ∈ T + [ m , ι ] is an irreducible and weakly symmetric tensor, we denote it as K ∈ WIT + [ m , ι ] , ρ ( K ) is the largest Z 1 -eigenvalue of K , by Proposition 4.9 in [33], the following lemma can be presented directly.

Lemma 6

Let K ∈ WIT + [ m , ι ] . Then ( ρ ( K ) , μ ) is a positive Z 1 -eigenpair of K .

Theorem 7

Let K ∈ WIT + [ m , ι ] , K ̲ s = ( k ̲ j j s ) = ( min j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ( k ^ j j 2 … j s … j m ) ) ∈ T [ 2 , ι ] , then

(16) ω ̲ 1 ≔ max s ∈ [ m ] \ { 1 } ( min j ∈ [ ι ] R j ( K ̲ s ) ) ≤ ρ ( K ) ≤ ω 1 ,

where

R j ( K ̲ s ) = ∑ j s ∈ [ ι ] k ̲ j j s .

Proof

The upper bound can be obtained directly by Theorem 2, we just prove the lower bound here. Let ( ρ ( K ) , μ ) be the positive Z 1 -eigenpair of K in Lemma 6 and μ = ( μ 1 , … , μ ι ) ⊤ , that is,

(17) K μ m − 1 = ρ ( K ) μ , ‖ μ ‖ 1 = 1 .

Let

μ p = min { μ j , j ∈ [ ι ] } .

Obviously, 1 ≥ μ p > 0 , we have

(18) ρ ( K ) μ p = ∑ j 2 , … , j m ∈ [ ι ] k p j 2 … j m μ j 2 … μ j m = ∑ j 2 , … , j m ∈ [ ι ] k ^ p j 2 … j m μ j 2 … μ j m .

Let t ∈ [ m ] \ { 1 } be an arbitrary index, we have

(19) ρ ( K ) μ p = ∑ j 2 , … , j m ∈ [ ι ] k p j 2 … j m μ j 2 … μ j m = ∑ j 2 , … , j m ∈ [ ι ] k ^ p j 2 … j m μ j 2 … μ j m

= ∑ j m ∈ [ ι ] ∑ j 2 , … , j m − 1 ∈ [ ι ] k ^ p j 2 … j m − 1 j m μ j 2 … μ j m − 1 μ j m ≥ min j m ∈ [ ι ] ∑ j 2 , … , j m − 1 ∈ [ ι ] k ^ p j 2 … j m − 1 j m μ j 2 … μ j m − 1 ∑ j m ∈ [ ι ] μ j m = min j m ∈ [ ι ] ∑ j 2 , … , j m − 1 ∈ [ ι ] k ^ p j 2 … j m − 1 j m μ j 2 … μ j m − 1 ⋮ ≥ min j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] k ^ p j 2 … j s … j m μ j s ≥ min j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] k ^ p j 2 … j s … j m μ p .

Therefore,

ρ ( K ) ≥ min j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] k ^ p j 2 … j s … j m = R p ( K ̲ s ) .

Since s ∈ [ m ] \ { 1 } is an arbitrary index, we obtain

□ ρ ( K ) ≥ max s ∈ [ m ] \ { 1 } ( min j ∈ [ ι ] R j ( K ̲ s ) ) .

Theorem 8

Let K ∈ WIT + [ m , ι ] , then

(20) ω ̲ 2 ≔ max s ∈ [ m ] \ { 1 } min j , t ∈ [ ι ] , j ≠ t 1 2 ( k ̲ j j + k ̲ t t + ( k ̲ j j − k ̲ t t ) 2 + 4 r j ( K ̲ s ) r t ( K ̲ s ) ) ≤ ρ ( K ) ≤ ω 2 ,

where

r j ( K ̲ s ) = R j ( K ̲ s ) − k ̲ j j .

Proof

The upper bound can be obtained directly by Theorem 3, we just prove the lower bound here. Let

μ p = min { μ j , j ∈ [ ι ] } , μ q = min { μ j , j ∈ [ ι ] , j ≠ p } .

Considering the p th equation in (17), we have

ρ ( K ) μ p ≥ min j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] k ^ p j 2 … j s … j m μ j s = min j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ( k ^ p j 2 … p … j m μ p ) + min j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] , j s ≠ p k ^ p j 2 … j s … j m μ j s ≥ min j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ( k ^ p j 2 … p … j m μ p ) + min j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] , j s ≠ p k ^ p j 2 … j s … j m μ q ,

which means

( ρ ( K ) − min j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ( k ^ p j 2 … p … j m ) ) μ p ≥ min j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ∑ j s ∈ [ ι ] , j s ≠ p k ^ p j 2 … j s … j m μ q .

Therefore,

(21) ( ρ ( K ) − k ̲ p p ) μ p ≥ ∑ j s ∈ [ ι ] , j s ≠ p k ̲ p j s μ q = r p ( K ̲ s ) μ q .

Similarly, considering the q th equation in (17), we have

(22) ( ρ ( K ) − k ̲ q q ) μ q ≥ ∑ j s ∈ [ ι ] , j s ≠ q k ̲ q j s μ p = r q ( K ̲ s ) μ p .

Multiplying (21) and (22), we obtain that

( ρ ( K ) − k ̲ p p ) ( ρ ( K ) − k ̲ q q ) ≤ r p ( K ̲ s ) r q ( K ̲ s ) ,

which means

ρ ( K ) ≥ 1 2 ( k ̲ p p + k ̲ q q + ( k ̲ p p − k ̲ q q ) 2 + 4 r p ( K ̲ s ) r q ( K ̲ s ) ) .

From the arbitrariness of t , we obtain

□ ρ ( K ) ≥ max s ∈ [ m ] \ { 1 } min j , t ∈ [ ι ] , j ≠ t 1 2 ( k ̲ j j + k ̲ t t + ( k ̲ j j − k ̲ t t ) 2 + 4 r j ( K ̲ s ) r t ( K ̲ s ) ) .

Theorem 9

Let K ∈ WIT + [ m , ι ] , S be a nonempty subset of  [ ι ] , S ¯ = [ ι ] \ S , and ι ≥ 2 , then

(23) ω ̲ 3 ≔ max s ∈ [ m ] \ { 1 } max S ⊆ [ ι ] min j ∈ S , t ∈ S ¯ 1 2 ( r j S ( K ̲ s ) + r t S ¯ ( K ̲ s ) + ( r j S ( K ̲ s ) − r t S ¯ ( K ̲ s ) ) 2 + 4 r j S ¯ ( K ̲ s ) r t S ( K ̲ s ) ) ≤ ρ ( K ) ≤ ω 3 ,

where

r j S ( K ̲ s ) = ∑ j s ∈ S k ̲ j j s , r j S ¯ ( K ̲ s ) = ∑ j s ∈ S ¯ k ̲ j j s , r t S ( K ̲ s ) = ∑ j s ∈ S k ̲ t j s , r t S ¯ ( K ̲ s ) = ∑ j s ∈ S ¯ k ̲ t j s .

Proof

The upper bound can be obtained directly by Theorem 4, and we just prove the lower bound here. Let

μ p = min { μ j , j ∈ S } , μ q = min { μ t , t ∈ S ¯ } .

Considering the p th equation in (17), we have

(24) ( ρ ( K ) − r p S ( K ̲ s ) ) μ p ≥ r p S ¯ ( K ̲ s ) μ q .

Considering the q th equation in (17), we have

(25) ( ρ ( K ) − r q S ¯ ( K ̲ s ) ) μ q ≥ r q S ( K ̲ s ) μ p .

Multiplying inequalities (24) and (25), we obtain that

( ρ ( K ) − r p S ( K ̲ s ) ) ( ρ ( K ) − r q S ¯ ( K ̲ s ) ) ≥ r p S ¯ ( K ̲ s ) r q S ( K ̲ s ) ,

which means

□ ρ ( K ) ≤ 1 2 ( r p S ( K ̲ s ) + r q S ¯ ( K ̲ s ) + ( r p S ( K ̲ s ) − r q S ¯ ( K ̲ s ) ) 2 + 4 r p S ¯ ( K ̲ s ) r q S ( K ̲ s ) ) .

The following relationships can be obtained by the technique in the proof of Theorem 5.

Theorem 10

Let K ∈ WIT + [ m , ι ] , then

(26) ω ̲ 3 ≥ ω ̲ 2 ≥ ω ̲ 1 .

Example 3

Let K = ( k j 1 j 2 j 3 i 4 ) ∈ T [ 4,2 ] with

k 1111 = 1 2 , k 2222 = 3 , k j 1 j 2 j 3 i 4 = 1 3

elsewhere. By computations, we have ρ ( K ) = 2.3519 . Different bounds for ρ ( K ) are shown in Table 2. We can see that the results in Theorem 9 are better than other bounds given in the literature.

Table 2

Different bounds for ρ ( K )

Theorem 3.3 of [26] ρ ( K ) ≤ 5.1934
Theorem 3.5 of [26] ρ ( K ) ≤ 3.2870
Theorem 2.1 of [38] ρ ( K ) ≤ 3.3333
Theorem 2.5 of [39] ρ ( K ) ≤ 3.3333
Theorem 7 0.6667 ≤ ρ ( K ) ≤ 3.3333
Theorems 8 and 9 0.6667 ≤ ρ ( K ) ≤ 3.0437

4 Determination of positive definiteness of even-order tensors

The Z -identity tensor ℐ Z = ( e j 1 … j m ) ∈ T [ m , ι ] is defined when m is even [15]:

e j 1 … j m = 1 m ! ∑ p ∈ ϕ m ε j p ( 1 ) j p ( 2 ) … ε j p ( m − 1 ) j p ( m ) ,

where ε represents the standard Kronecker delta.

Theorem 11

Let K = ( k j 1 j 2 … j m ) ∈ T [ m , ι ] be an even order tensor, τ ∈ σ Z 1 ( K ) , α j be an arbitrary real number for all j ∈ [ ι ] ,

ℬ ¯ s = ( b ¯ j j s ) = ( max j 2 , … , j s − 1 , j s + 1 , … , j m ∈ [ ι ] ( ∣ k ^ j j 2 … j s … j m − α j e j j 2 … j s … j m ∣ ) ) ∈ T [ 2 , ι ] .

Then,

∣ τ − α j ∣ ≤ R j ( ℬ ¯ s ) , s ∈ [ m ] \ { 1 } ,

where

R j ( ℬ ¯ s ) = ∑ j s ∈ [ ι ] b ¯ j j s .

Proof

Let

∣ μ p ∣ = max { ∣ μ j ∣ , j ∈ [ ι ] } .

By (2), we have

(27) ( τ − α p ) μ p = ∑ j 2 , … , j m ∈ [ ι ] ( k p j 2 … j m − α p e p j 2 … j m ) μ j 2 … μ j m = ∑ j 2 , … , j m ∈ [ ι ] ( k ^ p j 2 … j m − α p e p j 2 … j m ) μ j 2 … μ j m .

From the proof of Theorem 2, we obtain

□ ∣ τ − α p ∣ ≤ R p ( ℬ ¯ s ) .

Theorem 12

Let K = ( k j 1 j 2 … j m ) ∈ T [ m , ι ] be an even order tensor, τ ∈ σ Z 1 ( K ) , α j , α t ∈ R for all j , t ∈ [ ι ] and j ≠ t . Then,

∣ τ − α j ∣ ∣ τ − α t ∣ ≤ R j ( ℬ ¯ s ) R t ( ℬ ¯ s ) , s ∈ [ m ] \ { 1 } .

Proof

Let

∣ μ p ∣ = max { ∣ μ j ∣ , j ∈ [ ι ] } , ∣ μ q ∣ = max { ∣ μ j ∣ , j ∈ [ ι ] , j ≠ p } .

Considering the p th equation in (2), we have

( τ − α p ) μ p = ∑ j 2 , … , j m ∈ [ ι ] ( k p j 2 … j m − α p e p j 2 … j m ) μ j 2 … μ j m = ∑ j 2 , … , j m ∈ [ ι ] ( k ^ p j 2 … j m − α p e p j 2 … j m ) μ j 2 … μ j m ,

which means

(28) ∣ τ − α p ∣ ∣ μ p ∣ ≤ R p ( ℬ ¯ s ) ∣ μ q ∣ .

Similarly, considering the q th equation in (2), we have

(29) ∣ τ − α q ∣ ∣ μ q ∣ ≤ R q ( ℬ ¯ s ) ∣ μ p ∣ .

If ∣ μ q ∣ ≠ 0 , then multiplying inequalities (28) and (29), we obtain that

∣ τ − α p ∣ ∣ τ − α q ∣ ≤ R p ( ℬ ¯ s ) R q ( ℬ ¯ s ) .

If ∣ μ q ∣ = 0 , then ∣ τ − α p ∣ = 0 , we still have

□ ∣ τ − α p ∣ ∣ τ − α q ∣ ≤ R p ( ℬ ¯ s ) R q ( ℬ ¯ s ) .

Theorem 13

Let S be a nonempty subset of [ ι ] , S ¯ = [ ι ] \ S , and ι ≥ 2 , K = ( k j 1 j 2 … j m ) ∈ T [ m , ι ] be an even order tensor, τ ∈ σ Z 1 ( K ) , α j , α t ∈ R , and j ∈ S , t ∈ S ¯ . Then,

∣ τ − α j ∣ ≤ R j S ( ℬ ¯ s ) , j ∈ [ ι ] , s ∈ [ m ] \ { 1 } ,

and

( ∣ τ − α j ∣ − R j S ( ℬ ¯ s ) ) ( ∣ τ − α t ∣ − R t S ¯ ( K ¯ s ) ) ≤ R j S ¯ ( ℬ ¯ s ) R t S ( K ¯ s ) ,

where

R j S ( ℬ ¯ s ) = ∑ j s ∈ S b ¯ j j s , j ∈ S , R t S ¯ ( ℬ ¯ s ) = ∑ j s ∈ S ¯ b ¯ t j s , t ∈ S ¯ .

Proof

Let

∣ μ p ∣ = max { ∣ μ j ∣ , j ∈ S } , ∣ μ q ∣ = max { ∣ μ t ∣ , t ∈ S ¯ } .

Considering the p th equation in (2), we have

(30) ( ∣ τ − α p ∣ − R p S ( ℬ ¯ s ) ) ∣ μ p ∣ ≤ R p S ¯ ( ℬ ¯ s ) ∣ μ q ∣ .

Considering the q th equation in (2), we have

(31) ( ∣ τ − α q ∣ − R q S ¯ ( K ¯ s ) ) ∣ μ q ∣ ≤ R q S ( K ¯ s ) ∣ μ p ∣ .

If ∣ μ q ∣ = 0 , by (30), we have

∣ τ − α p ∣ ≤ R p S ( ℬ ¯ s ) .

If ∣ μ p ∣ = 0 , by (31), we have

∣ τ − α q ∣ ≤ R q S ¯ ( K ¯ s ) .

If ∣ μ p ∣ > 0 and ∣ μ q ∣ > 0 , multiplying inequalities (30) and (31), we obtain

□ ( ∣ τ − α p ∣ − R p S ( ℬ ¯ s ) ) ( ∣ τ − α q ∣ − R q S ¯ ( K ¯ s ) ) ≤ R p S ¯ ( ℬ ¯ s ) R q S ( K ¯ s ) .

Remark 6

Let α j = 0 for all j ∈ [ ι ] , then the results in Theorems 11, 12, and 13 reduce to the results in Theorems 2, 3, and 4.

Now, based on the inclusion sets introduced in this section, sufficient criteria for the positivity of tensors are first investigated as follows.

Theorem 14

Let K ∈ ST [ m , ι ] be an even order tensor, if there exists one s ∈ [ m ] \ { 1 } such that

α j > R j ( ℬ ¯ s ) ,

for all j ∈ [ ι ] , then K is a positive definite tensor.

Proof

By the results in Theorem 11, there is j 0 ∈ [ ι ] such that

(32) ∣ τ − α j 0 ∣ ≤ R j 0 ( ℬ ¯ s ) .

If τ ≤ 0 , and α j 0 > R j 0 ( ℬ ¯ s ) > 0 for all j 0 ∈ [ ι ] , we can obtain

∣ τ − α j 0 ∣ ≥ α j 0 > R j 0 ( ℬ ¯ s ) ,

which conflicts with the result (32). Therefore, τ > 0 and K should be a positive definite tensor.□

Theorem 15

Let K = ( k j 1 j 2 … j m ) ∈ ST [ m , ι ] be an even order tensor, if there exists one s ∈ [ m ] \ { 1 } such that

α j α t > R j ( ℬ ¯ s ) R t ( ℬ ¯ s ) ,

for all j , t ∈ [ ι ] , j ≠ t , then K is a positive definite tensor.

Proof

By the results in Theorem 12, there are j 0 , t 0 ∈ [ ι ] , j 0 ≠ t 0 such that

(33) ∣ τ − α j 0 ∣ ∣ τ − α t 0 ∣ ≤ R j 0 ( ℬ ¯ s ) R t 0 ( ℬ ¯ s ) .

If τ ≤ 0 ,

∣ τ − α j 0 ∣ ∣ τ − α t 0 ∣ ≥ α j 0 α t 0 > R j 0 ( ℬ ¯ s ) R t 0 ( ℬ ¯ s ) ,

which conflicts with the result (33). Therefore, τ > 0 and K should be a positive definite tensor.□

Theorem 16

Let K = ( k j 1 j 2 … j m ) ∈ T [ m , ι ] be an even order tensor, if there exists one s ∈ [ m ] \ { 1 } such that

α j > R j S ( ℬ ¯ s ) , f o r a t l e a s t o n e j ∈ S a n d ( α j − R j S ( ℬ ¯ s ) ) ( α t − R t S ¯ ( ℬ ¯ s ) ) > R j S ¯ ( ℬ ¯ s ) R t S ( K ¯ s ) ,

for all j ∈ S , t ∈ S ¯ , then K is a positive definite tensor.

Proof

By the results in Theorem 13, there is j 0 ∈ S , t 0 ∈ S ¯ such that

(34) ∣ τ − α j 0 ∣ ≤ R j 0 S ( ℬ ¯ s ) , j 0 ∈ [ ι ] ,

and

(35) ( ∣ τ − α j 0 ∣ − R j 0 S ( ℬ ¯ s ) ) ( ∣ τ − α t 0 ∣ − R t 0 S ¯ ( K ¯ s ) ) ≤ R j 0 S ¯ ( ℬ ¯ s ) R t 0 S ( K ¯ s ) .

If τ ≤ 0 , and α j 0 , α t 0 > 0 for all j 0 , t 0 ∈ [ ι ] , we obtain

∣ τ − α j 0 ∣ ≥ α j 0 > R j 0 S ( ℬ ¯ s )

and

( ∣ τ − α j 0 ∣ − R j 0 S ( ℬ ¯ s ) ) ( ∣ τ − α t 0 ∣ − R t 0 S ¯ ( K ¯ s ) ) ≥ ( α j 0 − R j 0 S ( ℬ ¯ s ) ) ( α t 0 − R t 0 S ¯ ( K ¯ s ) ) > R j 0 S ¯ ( ℬ ¯ s ) R t 0 S ( K ¯ s ) ,

which conflicts with the results (34) and (35). Therefore, τ > 0 and K should be a positive definite tensor.□

Example 4

Let K = ( k j 1 j 2 j 3 j 4 ) ∈ T [ 4,2 ] be a symmetric tensor with

k 1111 = 4 , k ϕ ( 1122 ) = 1 , k 2222 = 7 , k ϕ ( 1222 ) = − 3 ,

and k j 1 j 2 j 3 i 4 = 0 elsewhere.

By direct computations, we obtain that

k 1111 = 4 < r 1 ( K ) = 6 , k 2222 = 7 < r 2 ( K ) = 12 .

Then, we cannot use Corollary 1 in [20] to determine the positiveness of K .

By Corollary 2 in [32], we obtain that

l 2 = 3 ∣ a 2211 ∣ − 3 ∣ a 2122 ∣ = 3 − 9 = − 6 < 0 .

Hence, the results in Corollary 2 [32] cannot be applied to determine the positiveness of K .

By Theorem 14, let s = 2 , α 1 = 6 , α 2 = 9 , we have

α 1 = 6 > R 1 ( ℬ ¯ s ) = 5 , α 2 = 9 > R 2 ( ℬ ¯ s ) = 6 .

By the results in Theorem 14, K is positive definite. In fact,

σ Z 1 ( K ) = { 0.542 } , σ Z 2 ( K ) = { 1.07 , 3.9924 , 4 , 9.5589 } , σ H ( K ) = { 2.0039 , 4 , 4.265 , 15.6886 } ,

where σ Z 2 ( K ) is the set which contains all the Z 2 -eigenvalues of K and σ H ( K ) is the set which contains all the H -eigenvalues of K . From the results in Theorem 1, K is positive definite.

5 Conclusion

The Geršhgorin theorem, Brauer theorem, and S-type theorem for Z 1 -eigenvalues of tensors are presented. Based on these Geršhgorin-type theorems for Z 1 -eigenvalues, several sufficient criteria for the positivity of even order tensors are first introduced. Permutations are used to obtain more refined inclusion sets, but it is hard to obtain the optimal choice of permutations, and we will study it in the future.

Acknowledgement

The authors express their sincere gratitude to the anonymous referees for their careful reading of the manuscript and valuable comments.

  1. Funding information: This work was supported by Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2020]094, [2022]017), Science and Technology Foundation of Guizhou Province, China (Qian Ke He Ji Chu ZK[2021]Yi Ban 014).

  2. Author contributions: All authors contributed equally to this work.

  3. Conflict of interest: The authors state that there is no conflict of interest.

  4. Data availability statement: Not applicable.

References

[1] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput. 40 (2005), 1302–1324, DOI: https://doi.org/10.1016/j.jsc.2005.05.007. 10.1016/j.jsc.2005.05.007Search in Google Scholar

[2] L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSAP’05: Proceeding of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing, 2005, pp. 129–132, DOI: https://doi.org/10.1109/CAMAP.2005.1574201. 10.1109/CAMAP.2005.1574201Search in Google Scholar

[3] K. C. Chang and T. Tan Zhang, On the uniqueness and non-uniqueness of the positive Z-eigenvector for transition probability tensors, J. Math. Anal. Appl. 408 (2013), 525–540, DOI: https://doi.org/10.1016/j.jmaa.2013.04.019. 10.1016/j.jmaa.2013.04.019Search in Google Scholar

[4] M. Che, A. Cichocki, and Y. Wei, Neural networks for computing best rank-one approximations of tensors and its applications, Neurocomputing 267 (2017), 114–133, DOI: https://doi.org/10.1016/j.neucom.2017.04.058. 10.1016/j.neucom.2017.04.058Search in Google Scholar

[5] N. K. Bose and A. R. Modaress, General procedure for multivariable polynomial positivity with control applications, IEEE Trans. Automat. Control AC21 (1976), 596–601, DOI: https://doi.org/10.1109/TAC.1976.1101356. 10.1109/TAC.1976.1101356Search in Google Scholar

[6] N. K. Bose and P. S. Kamat, Algorithm for stability test of multidimensional filters, IEEE Trans. Acoust. Speech Signal Process. 20 (1975), 169–175, DOI: https://doi.org/10.1109/TASSP.1974.1162592. 10.1109/TASSP.1974.1162592Search in Google Scholar

[7] M. Fu, Comments on ‘A procedure for the positive definiteness of forms of even-order’, IEEE Trans. Autom. Control 43 (1998), 1430, DOI: https://doi.org/10.1109/9.720500. 10.1109/9.720500Search in Google Scholar

[8] M. A. Hasan and A. A. Hasan, A procedure for the positive definiteness of forms of even-order, IEEE Trans. Autom. Control 41 (1996), 615–617, DOI: https://doi.org/10.1109/9.489287. 10.1109/9.489287Search in Google Scholar

[9] W. Ku, Explicit criterion for the positive definiteness of a general quartic form, IEEE Trans. Autom. Control 10 (1965), 372–373, DOI: https://doi.org/10.1109/TAC.1965.1098177. 10.1109/TAC.1965.1098177Search in Google Scholar

[10] F. Wang and L. Qi, Comments on Explicit criterion for the positive definiteness of a general quartic form, IEEE Trans. Autom. Control 50 (2005), 416–418, DOI: https://doi.org/10.1109/TAC.2005.843851. 10.1109/TAC.2005.843851Search in Google Scholar

[11] J. Xie and A. Chang, On the Z-eigenvalues of the adjacency tensors for uniform hypergraphs, Linear Algebra Appl. 439 (2013), 2195–2204, DOI: https://doi.org/10.1016/j.laa.2013.07.016. 10.1016/j.laa.2013.07.016Search in Google Scholar

[12] J. Xie and A. Chang, On the Z-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph, Numer. Linear Algebra Appl. 20 (2013), 1030–1045, DOI: https://doi.org/10.1002/nla.1910. 10.1002/nla.1910Search in Google Scholar

[13] C. Bu, Y. Fan, and J. Zhou, Laplacian and signless Laplacian Z-eigenvalues of uniform hypergraphs, Front. Math. China 11 (2016), 511–520, DOI: https://doi.org/10.1007/s11464-015-0467-x. 10.1007/s11464-015-0467-xSearch in Google Scholar

[14] J. He, Y. Liu, J. Tian, and X. Liu, Upper bounds for signless Laplacian Z-spectral radius of uniform hypergraphs, Front. Math. China 14 (2019), 17–24, DOI: https://doi.org/10.1007/s11464-019-0743-2. 10.1007/s11464-019-0743-2Search in Google Scholar

[15] T. Kolda and J. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl. 32 (2011), 1095–1124, DOI: https://doi.org/10.1137/100801482. 10.1137/100801482Search in Google Scholar

[16] L. Chen, L. Han, and L. Zhou, Computing tensor eigenvalues via homotopy methods, SIAM J. Matrix Anal. Appl. 37 (2016), 290, DOI: https://doi.org/10.1137/15M1010725. 10.1137/15M1010725Search in Google Scholar

[17] C. Bu, Y. P. Wei, L. Sun, and J. Zhou, Brualdi-type eigenvalue inclusion sets of tensors, Linear Algebra Appl. 480 (2015), 168–175, DOI: https://doi.org/10.1016/j.laa.2015.04.034. 10.1016/j.laa.2015.04.034Search in Google Scholar

[18] C. Bu, X. Jin, H. Li, and C. Deng, Brauer-type eigenvalue inclusion sets and the spectral radius of tensors, Linear Algebra Appl. 512 (2017), 234–248, DOI: https://doi.org/10.1016/j.laa.2016.09.041. 10.1016/j.laa.2016.09.041Search in Google Scholar

[19] C. Li, Y. Li, and X. Kong, New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl. 21 (2014), 39–50, DOI: https://doi.org/10.1002/nla.1858. 10.1002/nla.1858Search in Google Scholar

[20] C. Li, A. Jiao, and Y. Li, An S-type eigenvalue localization set for tensors, Linear Algebra Appl. 493 (2016), 469–483, DOI: https://doi.org/10.1016/j.laa.2015.12.018. 10.1016/j.laa.2015.12.018Search in Google Scholar

[21] C. Li, F. Wang, J. Zhao, Y. Zhu, and Y. Li, Criterions for the positive definiteness of real supersymmetric tensors, J. Comput. Appl. Math. 255 (2014), 1–14, DOI: https://doi.org/10.1016/j.cam.2013.04.022. 10.1016/j.cam.2013.04.022Search in Google Scholar

[22] C. Deng, H. Li, and C. Bu, Brauer-type eigenvalue inclusion sets of stochastic/irreducible tensors and positive definiteness of tensors, Linear Algebra Appl. 556 (2018), 55–69, DOI: https://doi.org/10.1016/j.laa.2018.06.032. 10.1016/j.laa.2018.06.032Search in Google Scholar

[23] M. Che, G. Li, L. Qi, and Y. Wei, Pseudo-spectra theory of tensors and tensor polynomial eigenvalue problems, Linear Algebra Appl. 533 (2017), 536–572, DOI: https://doi.org/10.1016/j.laa.2017.07.026. 10.1016/j.laa.2017.07.026Search in Google Scholar

[24] G. Wang, G. Zhou, and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 187–198, DOI: https://doi.org/10.3934/dcdsb.2017009. 10.3934/dcdsb.2017009Search in Google Scholar

[25] Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl. 34 (2013), 1581–1595, DOI: https://doi.org/10.1137/130909135. 10.1137/130909135Search in Google Scholar

[26] W. Li, D. Liu, and S. Vong, Z-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl. 483 (2015), 182–199, DOI: https://doi.org/10.1016/j.laa.2015.05.033. 10.1016/j.laa.2015.05.033Search in Google Scholar

[27] J. He, Y. Liu, and G. Xu, Z-eigenvalues-based sufficient conditions for the positive deffiniteness of fourth-order tensors, Bull. Malays. Math. Sci. Soc. 43 (2020), 1069–1093, DOI: https://doi.org/10.1007/s40840-019-00727-7. 10.1007/s40840-019-00727-7Search in Google Scholar

[28] J. He, Y.-M. Liu, H. Ke, J.-K. Tian, and X. Li, Bounds for the Z-spectral radius of nonnegative tensors, SpringerPlus 5 (2016), 1727, 8, DOI: https://doi.org/10.1186/s40064-016-3338-3. 10.1186/s40064-016-3338-3Search in Google Scholar PubMed PubMed Central

[29] J. He and T.-Z. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Appl. Math. Lett. 38 (2014), 110–114, DOI: https://doi.org/10.1016/j.aml.2014.07.012. 10.1016/j.aml.2014.07.012Search in Google Scholar

[30] Q. Liu and Y. Li, Bounds for the Z-eigenpair of general nonnegative tensors, Open Math. 14 (2016), 181–194, DOI: https://doi.org/10.1515/math-2016-0017. 10.1515/math-2016-0017Search in Google Scholar

[31] J. Zhao, A new Z-eigenvalue localization set for tensors, J. Inequal. Appl. 2017 (2017), 85, DOI: https://doi.org/10.1186/s13660-017-1363-6. 10.1186/s13660-017-1363-6Search in Google Scholar PubMed PubMed Central

[32] J. Zhao and C. Sang, Z-Eigenvalue intervals of even-order tensors with application to judge the strong ellipticity of an elasticity tensor, Acta Appl. Math. 182 (2022), no. 1, 5, DOI: https://doi.org/10.1007/s10440-022-00540-8. 10.1007/s10440-022-00540-8Search in Google Scholar

[33] K. C. Chang, K. J. Pearson, and T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl. 438 (2013), 4166–4182, DOI: https://doi.org/10.1016/j.laa.2013.02.013. 10.1016/j.laa.2013.02.013Search in Google Scholar

[34] C. Sang, A new Brauer-type Z-eigenvalue inclusion set for tensors, Numer. Algorithms 80 (2019), 781–794, DOI: https://doi.org/10.1007/s11075-018-0506-2. 10.1007/s11075-018-0506-2Search in Google Scholar

[35] C. Sang and Z. Chen, Optimal Z-eigenvalue inclusion intervals of tensors and their applications, J. Ind. Manag. Optim. 18 (2022)2435–2468, DOI: https://doi.org/10.3934/jimo.2021075. 10.3934/jimo.2021075Search in Google Scholar

[36] W. Li, D. Liu, M. K. Ng, and S. Vong, The uniqueness of multilinear PageRank vectors, Numer. Linear Algebra Appl. 24 (2017), e2107, DOI: https://doi.org/10.1002/nla.2107. 10.1002/nla.2107Search in Google Scholar

[37] C. J. Hillar and L.-H. Lim, Most tensor problems are NP-hard, J. ACM 60 (2013), 1–39, DOI: https://doi.org/10.1145/2512329. 10.1145/2512329Search in Google Scholar

[38] Y. Liu, C. Li, and Y. Li, A refined bound for the Z1-spectral radius of tensors, Filomat 3 (2020), 2123–2129, DOI: https://doi.org/10.2298/FIL2007123L. 10.2298/FIL2007123LSearch in Google Scholar

[39] G. Wang, C. Deng, and C. Bu, Some upper bounds on Zt-eigenvalues of tensors, Appl. Math. Comput. 329 (2018), 266–277, DOI: https://doi.org/10.1016/j.amc.2018.01.064. 10.1016/j.amc.2018.01.064Search in Google Scholar
Received: 2024-09-16
Revised: 2025-02-06
Accepted: 2025-03-10
Published Online: 2025-04-01

© 2025 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/dema-2025-0117
Keywords for this article
H-eigenvalue; Geršhgorin theorem; Z 1-eigenvalue; Z 2-eigenvalue; bounds
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
  3. Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
  4. Grand Triebel-Lizorkin-Morrey spaces
  5. Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order
  6. Proofs of two conjectures involving sums of normalized Narayana numbers
  7. On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth
  8. New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations
  9. Some inequalities for rational function with prescribed poles and restricted zeros
  10. Lucas difference sequence spaces defined by Orlicz function in 2-normed spaces
  11. Evaluating the efficacy of fuzzy Bayesian networks for financial risk assessment
  12. Fixed point results for contractions of polynomial type
  13. Estimation for spatial semi-functional partial linear regression model with missing response at random
  14. Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
  15. New forms of bilateral inequalities for K-g-frames
  16. Rate of pole detection using Padé approximants to polynomial expansions
  17. Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth
  18. Note on the shape-preservation of a new class of Kantorovich-type operators via divided differences
  19. Geršhgorin-type theorems for Z1-eigenvalues of tensors with applications
  20. New topologies derived from the old one via operators
  21. Blow up solutions for two-dimensional semilinear elliptic problem of Liouville type with nonlinear gradient terms
  22. Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity
  23. Nonparametric expectile shortfall regression for functional data
  24. Advancing analytical solutions: Novel wave insights and methodologies for beta fractional Kuralay-II equations
  25. A generalized p-Laplacian problem with parameters
  26. A study of solutions for several classes of systems of complex nonlinear partial differential difference equations in ℂ2
  27. Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
  28. ω ℒ -biprojective and ω ¯ -contractible Banach algebras
  29. Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function
  30. Solutions of several general quadratic partial differential-difference equations in ℂ2
  31. Inequalities for the generalized trigonometric functions with respect to weighted power mean
  32. Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions
  33. Hankel determinants for q-starlike functions connected with q-sine function
  34. System of partial differential hemivariational inequalities involving nonlocal boundary conditions
  35. A new family of multivalent functions defined by certain forms of the quantum integral operator
  36. A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications
  37. Weighted composition operators on bicomplex Lorentz spaces with their characterization and properties
  38. Behavior of spatial curves under different transformations in Euclidean 4-space
  39. Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
  40. A new kind of Durrmeyer-Stancu-type operators
  41. A study of generalized Mittag-Leffler-type function of arbitrary order
  42. On the approximation of Kantorovich-type Szàsz-Charlier operators
  43. Split quaternion Fourier transforms for two-dimensional real invariant field
  44. Review Article
  45. Characterization generalized derivations of tensor products of nonassociative algebras
  46. Special Issue on Differential Equations and Numerical Analysis - Part II
  47. Existence and optimal control of Hilfer fractional evolution equations
  48. Persistence of a unique periodic wave train in convecting shallow water fluid
  49. Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities
  50. Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives
  51. Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
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  57. Some fixed point results with the vector degree of nondensifiability in generalized Banach spaces and application on coupled Caputo fractional delay differential equations
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  60. Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions
  61. Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
  62. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
  63. Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping
  64. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part II
  65. Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
  66. New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
  67. Special Issue on Approximation Theory and Special Functions 2024 conference
  68. Ulam-type stability for Caputo-type fractional delay differential equations
  69. Faster approximation to multivariate functions by combined Bernstein-Taylor operators
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  71. Some special functions and cylindrical diffusion equation on α-time scale
  72. (q, p)-Mixing Bloch maps
  73. Orthogonalizing q-Bernoulli polynomials
  74. On better approximation order for the max-product Meyer-König and Zeller operator
  75. Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
  76. Special Issue on Variational Methods and Nonlinear PDEs
  77. A note on mean field type equations
  78. Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
  79. Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
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