Two new eigenvalue localization sets for tensors and theirs applications

Definition 1.1

([1, 3]). Let 𝓐 = (a_{i1⋯ im}) ∈ ℂ^[m,n]. A pair (λ, x) ∈ ℂ × (ℂⁿ ∖ {0}) is called an eigenvalue-eigenvector (or simply eigenpair) of 𝓐 if

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

Definition 1.2

([1, 3]). Let 𝓐 = (a_{i1⋯ im}) ∈ ℝ^[m,n]. A pair (λ, x) ∈ ℂ × (ℂⁿ ∖ {0}) is called an E-eigenpair of 𝓐 if

(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

It is shown in [1] that a real even-order symmetric tensor 𝓐 = (a_{i1⋯ im}) 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, 4–7].

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 [8–19].

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 = (a_{i1⋯ im}) ∈ ℂ^[m,n]. Then

where σ(𝓐) is the set of all eigenvalues of 𝓐 and

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 = (a_{i1⋯ im}) ∈ ℂ^[m,n]. Then

where

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 = (a_{i1⋯ im}) ∈ ℂ^[m,n], S be a nonempty proper subset of N. Then

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 = (a_{i1⋯ im}) ∈ ℝ^[m,n]. Then

where

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.

Theorem 2.1

Let 𝓐 = (a_{i1⋯ im}) ∈ ℂ^[m,n]. Then

where

Proof

Let λ be an eigenvalue of 𝓐 with corresponding eigenvector x = (x₁, ⋯, x_n)^T ∈ ℂⁿ ∖ {0}, i.e.,

Let |x_p| = max{|x_i| : i ∈ N}. Then, |x_p| > 0. From (1), we have

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

equivalently,

If |x_j| = 0, by |x_p| > 0, we have |λ−ap⋯p|−rpj(A)≤0. Then

which implies that λ ∈ 𝓚_p,j(𝓐) ⊆ 𝓚^∩(𝓐). otherwise, |x_j| > 0. Similarly, from (1), we can obtain

Multiplying (2) with (3) and noting that |x_p|^m−1|x_j|^m−1 > 0, we have

then λ ∈ 𝓚_p,j(𝓐) ⊆ 𝓚^∩(𝓐) . From the arbitrariness of j, we have λ∈⋂j∈N,j≠pKp,j(A). Furthermore, λ ∈ ⋃i∈N⋂j∈N,j≠iKi,j(A). □

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

Theorem 2.2

Let 𝓐 = (a_{i1⋯ im}) ∈ ℂ^[m,n], S be a nonempty proper subset of N. Then

Proof

Let S be the complement of S in N. According to Theorem 2.3 in [4], 𝓚^S(𝓐) ⊆ 𝓚(𝓐) ⊆Γ(𝓐) . Hence, we only prove 𝓚^∩(𝓐) ⊆ 𝓚^S(𝓐) . Let z ∈ 𝓚^∩(𝓐), then there exists i₀ ∈ N, such that z ∈ 𝓚_i0,j(𝓐), ∀ j ∈ N, j ≠ i₀. If i₀ ∈ S, then for any j ∈S, we have z∈⋃i0∈S,j∈S¯Ki0,j(A)⊆KS(A). If i₀ ∈S, then for any j ∈ S, we have z∈⋃i0∈S¯,j∈SKi,j(A)⊆KS(A). 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 𝓚^S(𝓐).

As shown in [1, 4–7], 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 = (a_{i1⋯ im}) ∈ ℝ^[m,n] be an even-order symmetric tensor with a_{k⋯ k} > 0 for all k ∈ N. If ∀ i ∈ N, ∃ j ∈ N, j ≠ i,

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 i₀ ∈ N, for any j ∈ N, j ≠ i₀,

Suppose that λ ≤ 0. Then for i₀ ∈ N, ∃ j₀, such that a_{i0⋯ i0} > 0, a_{j0⋯ j0} > 0, and

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 = (a_{i1⋯ im}) ∈ ℝ^[m,n] be an even-order symmetric tensor with a_{k⋯ k} ≥ 0 for all k ∈ N. If ∀ i ∈ N, ∃ j ∈ N, j ≠ i,

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 = (a_ijkl) ∈ ℝ^[4,3] be a symmetric tensor with elements defined as follows:

By computations, we get that

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.

Theorem 3.1

Let 𝓐 = (a_{i1⋯ im}) ∈ ℝ^[m,n], S be a nonempty proper subset of N, S be the complement of S in N. Then

where σ_E(𝓐) is the set of all E-eigenvalues of 𝓐 and

Proof

Let λ be an E-eigenvalue of 𝓐 with corresponding eigenvector x = (x₁,⋯,x_n)^T ∈ ℂⁿ ∖ {0}, i.e.,

Let |x_p| = max{|x_i| : i ∈ S} and |x_q| = max{|x_j| : j ∈ S}. Then, at least one of |x_p| and |x_q| is nonzero. We next distinguish two cases to prove.

Case I. Suppose that |xq|≥|xp|, then |xq|=maxj∈N|xj| and 0<|xq|m−1≤|xq|≤1. From (8), we have

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

i.e.,

If |x_p| = 0, by |x_q| > 0, we have |λ|−RqΔS¯(A)≤0. Then

which implies that λ∈Ωp,qS(A)⊆ΩS(A). If |x_p| > 0, from (8), we can obtain

Multiplying (9) with (10) and noting that |x_p|^m−1|x_q|^m−1 > 0, we have

which leads to λ∈Ωp,qS(A)⊆ΩS(A).

Case II. Suppose that |x_p| ≥ |x_q|, then |xp|=maxi∈N|xi| and 0 < |x_p|^m−1 ≤ |x_p| ≤ 1. Similar to (9), we can obtain

If |x_q| = 0, by |x_p| > 0, we have |λ|−RpΔs¯¯(A)≤0. Then

which implies that λ∈Ωq,pS¯(A)⊆ΩS(A). If |x_q| > 0, similar to (10), we have

Multiplying (11) with (12) and noting that |x_p|^m−1|x_q|^m−1 > 0, we have

which leads to λ∈Ωq,pS¯(A)⊆ΩS¯(A). The conclusion follows from Cases I and II. □

Theorem 3.2

Let 𝓐 = (a_{i1⋯ im}) ∈ ℝ^[m,n], S be a nonempty proper subset of N, S be the complement of S in N. Then

Proof

Let λ ∈ Ω^S(𝓐) . Then

Without loss of generality, suppose that λ∈⋃i∈S,j∈S¯Ωi,jS(A) (we can prove it similarly if λ∈⋃i∈S¯,j∈SΩi,jS¯(A)).

Then there are i ∈ S and j ∈S such that λ∈Ωi,jS(A) , i.e.,

If Rj(A)RjΔS(A)=0, then λ=0 or |λ|≤RjΔS¯(A)≤Rj(A). Hence, λ ∈ R_i (𝓐) ⋃ R_j(𝓐). otherwise, from (13), we have

Furthermore,

or

which implies that λ ∈ R_i (𝓐)⋃ R_j(𝓐) . □

Remark 3.3

From Theorem 3.2, we known that the set Ω^S(𝓐) in Theorem 3.1 localizes all E-eigenvalues of a tensor 𝓐 more precisely than the set Γ^(A) 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 = (a_{i1⋯ im}) ∈ 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

where

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) ∈ Ω^S(𝓐) . Then

We next distinguish two cases to prove.

Case I: If ϱ(A)∈⋃i∈S,j∈S¯Ωi,jS(A), then there exists i ∈ S, j ∈S, such that

Then

Furthermore,

Case II: If ϱ(A)∈⋃i∈S¯,j∈SΩi,jS¯(A), similar to the proof of Case I, we can obtain

The conclusion follows from Cases I and II. □

Theorem 3.5

Let 𝓐 = (a_{i1⋯ im}) ∈ 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

Proof

Here, we only prove that when ΨS(A)=maxi∈S,j∈S¯ΨijS(A), (14) holds. Similarly, we can also prove that (14) holds if ΨS(A)=maxi∈S¯,j∈SΨijS¯(A). Next, we divide two cases to prove.

Case I: For any i ∈ S,j ∈S, if R_i(𝓐) ≤ R_j(𝓐), then

Case II: For any i ∈ S,j ∈S, if R_j(𝓐) ≥ R_i(𝓐), then 0 ≤ RjΔS(A)≤Ri(A)−RjΔS¯(A), and

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 [8–13] in some cases by the following example.

Example 3.7

Let 𝓐 = (a_ijk) ∈ ℝ^[3,3] with entries defined as follows:

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

By Theorem 3.5 of [11], we have

By Theorem 4.6 of [8], we have

By Theorem 4.7 of [8], we have

By Theorem 4.5 of [8] and Theorem 6 of [12], we both have

By Theorem 2.9 of [13], we have

Let S = {1},S = {2,3}. By Theorem 3.4, we obtain

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