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Open Mathematics

formerly Central European Journal of Mathematics

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

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Volume 14, Issue 1

Issues

New iterative codes for 𝓗-tensors and an application

Feng Wang / Deshu Sun
Published Online: 2016-04-23 | DOI: https://doi.org/10.1515/math-2016-0022

Abstract

New iterative codes for identifying 𝓗 -tensor are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor, i.e., an even-degree homogeneous polynomial form are given. Advantages of results obtained are illustrated by numerical examples.

Keywords: 𝓗-tensors; Symmetric tensors; Homogeneous polynomial; Positive definiteness; Irreducible

MSC 2010: 15A18; 15A21; 15A69

1 Introduction

Let C(R) be the complex(real) field and N = {1, 2, ... n}. We call 𝒜 = {ai1i2...im) an m-order n-dimensional complex(real) tensor, if ai1i2⋯im∈C(R), where ij = 1, ..., n for j = 1, ..., m [1–3]. A tensor 𝒜 = (ai1i2...im) is called symmetric [4], if ai1i2⋯im=aπ(i1i2⋯im),   ∀π∈Πm, where Πm is the permutation group of m indices. Furthermore, an m-order n-dimensional tensor 𝓘 = (δi1i2...im) is called the unit tensor [5], if its entries δi1i2⋯im={1, if  i1=⋯=im, 0, otherwise.

Consider the following positive definiteness identification problem [1].

For a real valued polynomial f, how to check whether f is positive definite, i.e., f(x)>0  for  any  x∈Rn,    x≠0, (1)

or not?

Problem 1.1 appears in numerous application domains [6–8]. In particular, the positive definiteness of multivariate polynomial f plays an important role in the stability study of nonlinear autonomous systems via Lyapunov’s direct method in automatic control [7, 9, 10], such as the multivariate network realizability theory [11], a test for Lyapunov stability in multivariate filters [12], a test of existence of periodic oscillations using Bendixon’s theorem [13], and the output feedback stabilization problems [14].

In 2005, Qi [1] defined the positive definiteness of a symmetric tensor 𝒜, i.e., we call 𝒜 = (ai1i2...im) positive definite if the following mth-degree homogeneous polynomial f(x) is positive definite:

f(x)=Axm=∑i1, i2, …, im∈Nai1i2⋯imxi1xi2...xim, (2)

where x = (x1, x2, .... xn) ∈ Rn Hence, we only research the positive definiteness of a symmetric tensor 𝒜 instead of f(x). Also in [1], Qi presented the concept of H-eigenvalues, and used it to verify the positive definiteness of an even-order symmetric tensor (see Proposition 1).

([1]). Let 𝒜 be an even-order real symmetric tensor, then 𝒜 is positive definite if and only if all of its H-eigenvalues are positive.

From Proposition 1.2, we can verify the positive definiteness of an even-order symmetric tensor 𝒜 (the positive definiteness of the mth-degree homogeneous polynomial f(x)) by computing the H-eigenvalues of 𝒜 . But it is not easy to compute all these H-eigenvalues when m and n are large. Recently, by introducing the definition of 𝓗-tensors [15, 16], Li et al. [16] provided a practical sufficient condition for identifying the positive definiteness of an even-order symmetric tensor (see Proposition 1. 2).

([16]). Let 𝒜 = (ai1i2...im) be an even-order real symmetric tensor with ak ... k > 0 for all k ∈ N. If 𝒜 is an 𝓗-tensor, then 𝒜 is positive definite.

Now, some definitions and notation are given, which will be used in the sequel.

([1]). Let 𝒜 = (ai1i2...im) be an m-order n-dimensional complex tensor. If |aii⋯i|≥∑i2, …, im∈Nδii2….im=0|aii2⋯im|, ∀i∈N.(3) then 𝒜 is called a diagonally dominant tensor; if all strict inequalities in (3) hold, then 𝒜 is called a strictly diagonally dominant tensor.

([15]). Let 𝒜 = 𝒜 = (ai1...im) be an m-order n-dimensional complex tensor. 𝒜 is called an 𝓗-tensor if there is a positive vector x = (x1, x2, . . . xn)T ∈ Rn such that |aii⋯i|xim−1>∑i2, …, im∈N, δii2⋯im=0|aii2⋯im| xi2⋯xim,             i=1, 2, ⋯, n.

([3]). Let 𝒜 = (ai1i2...im) be an m-order n-dimensional complex tensor, X = diag (x1, x2, ... xn). Denote B=(bi1⋯im)=AXm−1,         bi1i2⋯im=ai1i2⋯imxi2xi3⋯xim,         ij∈N,         j∈N, then 𝓑 is called the product of the tensor 𝒜 and the matrix X.

([5]). An m-order n-dimensional complex tensor 𝒜 = (ai1...im) is called reducible, if there exists a nonempty proper index subset I ⊂ N such that ai1i2⋯im=0,         ∀i1∈I,         ∀i2, …, im∉I.

If 𝒜 is not reducible, then we call 𝒜 irreducible.

Let S be a nonempty subset of N and let N\S be the complement of S in N . Given an m-order n-dimensional complex tensor 𝒜 = (ai1i2...im), we denote Ri(A)=∑i2, …, im∈Nδii2…im=0aii2⋯im|=∑i2, …, im∈N|aii2⋯im|−|aii⋯i|, N1={i∈N:0<|aii⋯i|=Ri(A)}, N2={i∈N:0<|aii⋯i|<Ri(A)}, N3={i∈N:|aii⋯i|>Ri(A)}, μi=Ri(A)−|aii⋯i|Ri(A), ∀i∈N2, r0=1, r1=maxi∈N3Ri(A)|aii⋯i|, σk+1, i=∑i2⋯im∈N0m−1|aii2⋯im|+∑i2⋯im∈N2m−1maxj∈{i2, ⋯, im}{μj}|aii2⋯im|+rk∑i2⋯im∈N3m−1δii2…im=0|aii2⋯im|, ∀i∈N3, k∈Z+={0, 1, 2, ⋯}, rk+1=maxi∈N3σk+1, i|aii⋯i|, k=1, 2, 3, ⋯, Sm−1={i2i3⋯im:ij∈S, j=2, 3, ⋯, m}, Nm−1∖Sm−1=i2i3⋯im:i2i3⋯im∈Nm−1andi2i3⋯im∉Sm−1, N0m−1=Nm−1∖(N2m−1∪N3m−1).

It is well known that if N1 ∪ N2 = ∅, then 𝒜 is an 𝓗-tensor, and if 𝒜 is an 𝓗-tensor, then N3 ∅ ; [16]. So we always assume that both N1 ∪ N2 and N3 are not empty. In addition, we also assume that 𝒜 satisfies: aii...i ≠ 0,, Ri(𝒜) ≠ 0, ∀i ∈ N.

This article is organized as follows: In Section 2, New iterative codes for identifying 𝒜-tensors are obtained. As an application, some new sufficient conditions of the positive definiteness for an even-order real symmetric tensor are presented in Section 3. Numerical examples are given to verify the corresponding results.

2 Criteria for identifying 𝒜-tensors

In this section, we give new iterative codes for identifying 𝒜-tensors.

([15]). If 𝒜 is a strictly diagonally dominant tensor, then 𝒜 is an 𝓗-tensor.

([16]). Let 𝒜 = (ai1...im) be an m-order n-dimensional complex tensor. If 𝒜 is irreducible, |aii⋯i|≥Ri(A),             ∀i∈N, and strictly inequality holds for at least one i, then 𝒜 is an 𝓗-tensor.

([3]). Let 𝒜 = (ai1...im) be an m-order n-dimensional complex tensor. If there exists a positive diagonal matrix X such that 𝒜Xm–1 is an 𝓗-tensor, then 𝒜 is an 𝓗-tensor.

Let 𝒜 = (ai1...im) be an m-order n-dimensional complex tensor. If there exists k ∈ N such that |aii⋯i|>∑i2⋯im∈N0m−1δii2…im=0|aii2⋯im|+∑i2⋯im∈N2m−1maxj∈{i2,⋯,im}μj|aii2⋯im|+∑i2⋯im∈N3m−1maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|,∀i∈N1,(4) and |aii⋯i|μi>∑i2⋯im∈N0m−1|aii2⋯im|+∑i2⋯im∈N2m−1δii2…im=0maxj∈{i2,⋯,im}μj|aii2⋯im|+∑i2⋯im∈N3m−1maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|,∀i∈N2,(5) then 𝒜 is an 𝓗-tensor.

For all i ∈ N1, we denote Mi=1∑i2⋯im∈N3m−1|aii2⋯im||aii⋯i|μi−∑i2⋯im∈N0m−1δii2…im=0|aii2⋯im|−∑i2⋯im∈N2m−1maxj∈{i2,⋯,im}μj|aii2⋯im|−∑i2⋯im∈N3m−1maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|,

and for all i ∈ N2, we denote Mi=1∑i2⋯im∈N3m−1|aii2⋯im||aii⋯i|μi−∑i2⋯im∈N0m−1|aii2⋯im|−∑i2⋯im∈N2m−1δii2…im=0maxj∈{i2,⋯,im}μj|aii2⋯im|−∑i2⋯im∈N3m−1maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|.

If ∑i2i3⋯im∈N3m−1|aii2⋯im|=0, we denote Mi = +∞. From Inequalities (4) and (5), we obtain Mi < 0(i ∈ N1 ⋃ N2). Hence, there exists a positive number ε such that 0<ε<minmini∈N1∪N2Mi,1−maxj∈N3σk+1,j|ajj⋯j|.(6)

Let the matrix X = diag(x1, x2, . . . ; xn), where xi=1, i∈N1, μi1m−1, i∈N2, ε+σk+1, i|aii⋯i|1m−1, i∈N3.

By Inequality (6), we have (ε+σk+1, i|aii⋯i|)1m−1<1(∀i∈N3). As ε ≠ +∞, xi ≠ +∞, which implies that X is a diagonal matrix with positive entries. Let 𝓑 = (bi1i2...im) = 𝒜Xm − 1. Next, we will prove that 𝓑 is strictly diagonally dominant.

First, we consider i ∈ N1. If ∑i2i3⋯im∈N3m−1|aii2⋯im|=0, then by Inequality (4), we have Ri(B)=∑i2i3⋯im∈N0m−1δii2⋯im=0|aii2⋯im|xi2⋯xim+∑i2i3⋯im∈N2m−1|aii2⋯im|(μi2)1m−1⋯(μim)1m−1≤∑i2i3⋯im∈N0m−1δii2⋯im=0|aii2⋯im|+∑i2i3⋯im∈N2m−1maxj∈{i2,⋯,im}μj|aii2⋯im|<|aii⋯i|=|bii⋯i|.(7)

If ∑i2i3⋯im∈N3m−1|aii2⋯im|≠0, then by Inequalities (4) and (6), we get Ri(B)=∑i2i3⋯im∈N0m−1δii2⋯im=0|aii2⋯im|xi2⋯xim+∑i2i3⋯im∈N2m−1|aii2⋯im|(μi2)1m−1⋯(μim)1m−1+∑i2i3⋯im∈N3m−1|aii2⋯im|ε+σk+1,i2|ai2i2⋯i2|1m−1⋯ε+σk+1,im|aimim⋯im|1m−1≤∑i2i3⋯im∈N0m−1δii2⋯im=0|aii2⋯im|+∑i2i3⋯im∈N2m−1maxj∈{i2,⋯,im}μj|aii2⋯im|+∑i2i3⋯im∈N3m−1|aii2⋯im|ε+maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j|=∑i2i3⋯im∈N0m−1δii2⋯im=0|aii2⋯im|+∑i2i3⋯im∈N2m−1maxj∈{i2,⋯,im}μj|aii2⋯im|+∑i2i3⋯im∈N3m−1maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|+ε∑i2i3⋯im∈N3m−1|aii2⋯im|<|aii⋯i|=|bii⋯i|.(8)

Next, we consider i ∈ N2. If ∑i2i3⋯im∈N3m−1|aii2⋯im|=0, then by Inequality (5), we have Ri(B)=∑i2i3⋯im∈N0m−1|aii2⋯im|xi2⋯xim+∑i2i3⋯im∈N2m−1δii2⋯im=0|aii2⋯im|(μi2)1m−1⋯(μim)1m−1≤∑i2i3⋯im∈N0m−1|aii2⋯im|+∑i2i3⋯im∈N2m−1δii2⋯im=0maxj∈{i2,⋯,im}μj|aii2⋯im|<|aii⋯i|μi=|bii⋯i|.(9)

If ∑i2i3⋯im∈N3m−1|aii2⋯im|≠0, then by Inequalities (5) and (6), we get Ri(B)=∑i2i3⋯im∈N0m−1|aii2⋯im|xi2⋯xim+∑i2i3⋯im∈N2m−1δii2⋯im=0|aii2⋯im|(μi2)1m−1⋯(μim)1m−1+∑i2i3⋯im∈N3m−1|aii2⋯im|ε+σk+1,i2|ai2i2⋯i2|1m−1⋯ε+σk+1,im|aimim⋯im|1m−1≤∑i2i3⋯im∈N0m−1|aii2⋯im|+∑i2i3⋯im∈N2m−1δii2⋯im=0maxj∈{i2,⋯,im}μj|aii2⋯im|+∑i2i3⋯im∈N3m−1|aii2⋯im|ε+maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j|=∑i2i3⋯im∈N0m−1|aii2⋯im|+μ∑i2i3⋯im∈N2m−1δii2⋯im=0|aii2⋯im|+∑i2i3⋯im∈N3m−1maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|+ε∑i2i3⋯im∈N3m−1|aii2⋯im|<|aii⋯i|μi=|bii⋯i|.(10)

Finally, we consider i ∈ N3. Since |aii... i| > Ri(𝒜), we have |aii⋯i|−∑i2i3⋯im∈N3m−1δii2⋯im=0|aii2⋯im|>0.(11)

And by σk+1, j|ajj⋯j|≤rk+1≤rk, ∀k∈N, j∈N3, we obtain ∑i2i3⋯im∈N0m−1|aii2⋯im|+∑i2i3⋯im∈N2m−1maxj∈{i2,⋯,im}μj|aii2⋯im|+∑i2i3⋯im∈N3m−1δii2⋯im=0maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|−σk+1,i=∑i2i3⋯im∈N2m−1δii2⋯im=0maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|−rk∑i2i3⋯im∈N2m−1δii2⋯im=0|aii2⋯im|≤0.(12)

By Inequalities (11), (12) and ε > 0, we get ε>1|aii⋯i|−∑i2i3⋯im∈N3m−1δii2⋯im=0|aii2⋯im|{∑i2i3⋯im∈N0m−1|aii2⋯im|+∑i2i3⋯im∈N2m−1maxj∈{i2,⋯,im}μj|aii2⋯im|+∑i2i3⋯im∈N3m−1δii2⋯im=0maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|−σk+1,i}.(13)

From Inequality (13), for any i ∈ N3, we have |bii⋯i|−Ri(B)=|aii⋯i|ε+σk+1,i|aii⋯i|−∑i2i3⋯im∈N0m−1|aii2⋯im|xi2⋯xim−∑i2i3⋯im∈N2m−1|aii2⋯im|(μi2)1m−1⋯(μim)1m−1−∑i2i3⋯im∈N3m−1δii2⋯im=0|aii2⋯im|ε+σk+1,i2|ai2i2⋯i2|1m−1⋯ε+σk+1,im|aimim⋯im|1m−1≥|aii⋯i|ε+σk+1,i|aii⋯i−∑i2i3⋯im∈N0m−1|aii2⋯im|−∑i2i3⋯im∈N2m−1maxj∈{i2,⋯,im}μj|aii2⋯im|−∑i2i3⋯im∈N3m−1δii2⋯im=0|aii2⋯im|ε+maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j|=ε(|aii⋯i|−∑i2i3⋯im∈N3m−1δii2⋯im=0|aii2⋯im|)+σk+1,i−∑i2i3⋯im∈N0m−1|aii2⋯im|−∑i2i3⋯im∈N2m−1maxj∈{i2,⋯,im}μj|aii2⋯im|−∑i2i3⋯im∈N3m−1δii2⋯im=0maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|>0.(14)

Therefore, from Inequalities (7-10) and (14), we obtain |bii... i| > Ri(𝓑) for all i ∈ N, and by Lemma 2.1, 𝓑 is an 𝓗-tensor. Furthermore, by Lemma 2.3, 𝒜 is an 𝓗-tensor. The proof is completed.

Let 𝒜 = (ai1 ...im) be an m-order n-dimensional complex tensor. If A is irreducible, and there exists k ∈ Z C such that |aii⋯i|≥∑i2⋯im∈N0m−1δii2…im=0|aii2⋯im|+∑i2⋯im∈N2m−1maxj∈{i2,⋯,im}μj|aii2⋯im|+∑i2⋯im∈N3m−1maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|,∀i∈N1,(15)

and |aii⋯i|μi≥∑i2⋯im∈N0m−1|aii2⋯im|+∑i2⋯im∈N2m−1δii2…im=0maxj∈{i2,⋯,im}μj|aii2⋯im|+∑i2⋯im∈N3m−1maxj∈{i2,i3,⋯,im}σk+1,j|ajj⋯j||aii2⋯im|,∀i∈N2,(16)

in addition, a strict inequality holds for at least one i ∈ N1 ⋃ N2, then 𝓐 is an 𝓗-tensor.

Let the matrix X = d iag (x1, x2, . . . ; xn), where xi={1, i∈N1, (μi)1m−1, i∈N2, (σk+1, i|aii⋯i|)1m−1, i∈N3.

By the irreducibility of 𝒜, we have xi ≠ +∞, then X is a diagonal matrix with positive diagonal entries. Let 𝓑 = (bi1...im) = 𝒜Xm −1.

Adopting the same procedure as in the proof of Theorem 2.4, we can obtain that |bii...i| ≥ Ri(𝓑)(∀i ∈ N), and there exists at least an i ∈ N1 ∪ N2 such that |bii...i| ≥ Ri (𝓑).

On the other hand, since 𝒜 is irreducible, then 𝓑 is also. Then by Lemma 2.2, we have that 𝓑 is an 𝓗-tensor. By Lemma 2.3, 𝒜 is an 𝓗-tensor. The proof is completed. □

Consider an 3-order 3-dimensional tensor 𝒜 = (aijk) defined as follows: A=[A(1, :, :), A(2, :, :), A(3, :, :)], ], [A(1, :, :)=(121011001110), A(2, :, :)=(1100180103), A(3, :, :)=(2000300015).

Obviously, |a111|=12, R1(A)=24, |a222|=18, R2(A)=6, |a333|=15, R3(A)=5,

so N1 = ∅, N2 = {1}, N3 = {2; 3}. By calculation, we have μ1=12, σ3, 2|a222|=37216, σ3, 3|a333|=19180.

Since ∑jk∈N02|a1jk|+∑jk∈N22δ1jk=0maxl∈{j,k}μl|a1jk|+∑jk∈N32maxl∈{j,k}σk+1,l|all⋯l||a1jk|=(1+0+1+1)+37216+10×37216+10×19180=1283216<6=|a11⋯1|μ1,

we know that A satisfies the conditions (k = 2) of Theorem 2.4, then 𝒜 is an 𝓗-tensor.

3 An application

In this section, based on the criteria of 𝓗-tensors in section 2, we present new conditions for identifying the positive definiteness of an even-order real symmetric tensor.

From Proposition 2, Theorem 2.4 and Theorem 2.5, we obtain easily the following result.

Let 𝒜 = ai1 ...im) be an m-order n-dimensional even-order real symmetric tensor with akk...k > 0 for all k ∈ N. If 𝒜 satisfies one of the following conditions, then 𝒜 is positive definite,

Let f(x)=Ax4=2x14+x24+30x34+30x44−8x1x33+8x2x43−12x33x4

be a 4th-degree homogeneous polynomial. We can get an 4-order 4-dimensional real symmetric tensor 𝒜 = (ai1i2i3i4), where a1111=2, a2222=1, a3333=30, a4444=30, a1333=a3133=a3313=a3331=−2, a2444=a4244=a4424=a4442=2, a3334=a3343=a3433=a4333=−3,

and other ai1i2i3i4 = 0. It can be verified that A satisfies all the conditions (k = 2) of Theorem 2.4. Therefore, from Theorem 3.1, we have that 𝒜 is positive definite, that is, f(x) is positive definite.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (11501141, 11361074), the Foundation of Science and Technology Department of Guizhou Province ([2015]2073, [2015]7206), the Natural Science Programs of Education Department of Guizhou Province ([2015]420), and the Research Foundation of Guizhou Minzu University (15XRY004).

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About the article

Received: 2015-11-21

Accepted: 2016-03-14

Published Online: 2016-04-23

Published in Print: 2016-01-01


Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 212–220, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0022.

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© 2016 Wang and Sun, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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