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.
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
where ij = 1, ..., n for j = 1, ..., m [1–3]. A tensor 𝒜 = (ai1i2...im) is called symmetric [4], if
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
Consider the following positive definiteness identification problem [1].
For a real valued polynomial f, how to check whether f is positive definite, i.e.,
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:
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
([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
([3]). Let 𝒜 = (ai1i2...im) be an m-order n-dimensional complex tensor, X = diag (x1, x2, ... xn). Denote
([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
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
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,
([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
For all i ∈ N1, we denote
and for all i ∈ N2, we denote
If
Let the matrix X = diag(x1, x2, . . . ; xn), where
By Inequality (6), we have
First, we consider i ∈ N1. If
If
Next, we consider i ∈ N2. If
If
Finally, we consider i ∈ N3. Since |aii... i| > Ri(𝒜), we have
And by
By Inequalities (11), (12) and ε > 0, we get
From Inequality (13), for any i ∈ N3, we have
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 ∈ ZCsuch that
and
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
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:
Obviously,
so N1 = ∅, N2 = {1}, N3 = {2; 3}. By calculation, we have
Since
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,
all the conditions of Theorem 2.4;
all the conditions of Theorem 2.5.
Let
be a 4th-degree homogeneous polynomial. We can get an 4-order 4-dimensional real symmetric tensor 𝒜 = (ai1i2i3i4), where
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).
References
[1] Qi, L.: Eigenvalues of a real supersymetric tensor. J. Symbolic Comput. 40 (2005), 1302-1324.10.1016/j.jsc.2005.05.007Search in Google Scholar
[2] Lim, L.H.: 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.Search in Google Scholar
[3] Kannana, M.R., Mondererb, N.S., Bermana, A.: Some properties of strong H-tensors and general H-tensors. Linear Algebra Appl. 476 (2015), 42-55.10.1016/j.laa.2015.02.034Search in Google Scholar
[4] Li, C. Q., Qi, L., Li, Y.T.: MB-tensors and MB0-tensors. Linear Algebra Appl. 484 (2015), 141-153..10.1016/j.laa.2015.06.030Search in Google Scholar
[5] Yang, Y., Yang, Q.: Fruther results for Perron-Frobenius theorem for nonnegative tensors. SIAM. J. Mayrix Anal. Appl. 31 (2010), 2517-2530.10.1137/090778766Search in Google Scholar
[6] Ni, Q., Qi, L., Wang, F.: An eigenvalue method for the positive definiteness identification problem. IEEE Trans. Automat. Control. 53 (2008), 1096-1107.10.1109/TAC.2008.923679Search in Google Scholar
[7] Ni, G., Qi, L., Wang, F., Wang, Y.: The degree of the E-characteristic polynomial of an even order tensor. J. Math. Anal. Appl. 329(2007), 1218-1229.10.1016/j.jmaa.2006.07.064Search in Google Scholar
[8] Qi, L., Teo, K.L.: Multivariate polynomial miniziation and its application in signal processing. J. Global Optimization. 26 (2003), 419-433.10.1023/A:1024778309049Search in Google Scholar
[9] Zhang, L., Qi, L., Zhou, G.: M-tensors and some applications. SIAM J. Matrix Anal. Appl. 32 (2014), 437-452.10.1137/130915339Search in Google Scholar
[10] Wang, F., Qi, L.: Comments on Explicit criterion for the positive definiteness of a general quartic form, IEEE Trans. Automat. Control. 50 (2005), 416-418.10.1109/TAC.2005.843851Search in Google Scholar
[11] Bose, N.K., Newcomb, R.W.: Tellegon’s theorem and multivariable realizability theory. Int. J. Electron. 36 (1974), 417-425.10.1080/00207217408900421Search in Google Scholar
[12] Bose, N.K., Kamat, P.S.: Algorithm for stability test of multidimensional filters. IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP20 (1975), 169-175.10.1109/TASSP.1974.1162592Search in Google Scholar
[13] Hsu, J.C., Meyer, A.U.: Mordern Control principles and Applications. McGrawHill, New York, 1968.Search in Google Scholar
[14] Anderson, B.D., Bose, N.K., Jury, E.I.: Output feddback stabilization and related problems-solutions via decision methods. IEEE Trans. Automat. Control. AC20 (1975), 55-66.10.1109/TAC.1975.1100846Search in Google Scholar
[15] Ding, W., Qi, L., Wei, Y.: M-tensors and nonsingular M-tensors. Linear Algebra Appl. 439 (2013), 3264-3278.10.1016/j.laa.2013.08.038Search in Google Scholar
[16] Li, C. Q., Wang, F., Zhao, J.X., Zhu, Y., Li, Y.T.: Criterions for the positive definiteness of real supersymmetric tensors. J. Comput. Appl. Math. 255 (2014), 1-14.10.1016/j.cam.2013.04.022Search in Google Scholar
© 2016 Wang and Sun, published by De Gruyter Open
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.