Consider the following positive definiteness identification problem .
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 , a test for Lyapunov stability in multivariate filters , a test of existence of periodic oscillations using Bendixon’s theorem , and the output feedback stabilization problems .
In 2005, Qi  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 , Qi presented the concept of H-eigenvalues, and used it to verify the positive definiteness of an even-order symmetric tensor (see Proposition 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.  provided a practical sufficient condition for identifying the positive definiteness of an even-order symmetric tensor (see Proposition 1. 2).
(). 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.
If 𝒜 is not reducible, then we call 𝒜 irreducible.
It is well known that if N1 ∪ N2 = ∅, then 𝒜 is an 𝓗-tensor, and if 𝒜 is an 𝓗-tensor, then N3 ∅ ; . 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.
(). If 𝒜 is a strictly diagonally dominant tensor, then 𝒜 is an 𝓗-tensor.
(). 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.
By Inequality (6), we have
in addition, a strict inequality holds for at least one i ∈ N1 ⋃ N2, then 𝓐 is an 𝓗-tensor.
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).
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.
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,
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.
This work was supported by the National Natural Science Foundation of China (11501141, 11361074), the Foundation of Science and Technology Department of Guizhou Province (2073, 7206), the Natural Science Programs of Education Department of Guizhou Province (420), and the Research Foundation of Guizhou Minzu University (15XRY004).
Qi, L.: Eigenvalues of a real supersymetric tensor. J. Symbolic Comput. 40 (2005), 1302-1324.
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.
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.
Li, C. Q., Qi, L., Li, Y.T.: MB-tensors and MB0-tensors. Linear Algebra Appl. 484 (2015), 141-153..
Yang, Y., Yang, Q.: Fruther results for Perron-Frobenius theorem for nonnegative tensors. SIAM. J. Mayrix Anal. Appl. 31 (2010), 2517-2530.
Ni, Q., Qi, L., Wang, F.: An eigenvalue method for the positive definiteness identification problem. IEEE Trans. Automat. Control. 53 (2008), 1096-1107.
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.
Qi, L., Teo, K.L.: Multivariate polynomial miniziation and its application in signal processing. J. Global Optimization. 26 (2003), 419-433.
Zhang, L., Qi, L., Zhou, G.: M-tensors and some applications. SIAM J. Matrix Anal. Appl. 32 (2014), 437-452.
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.
Bose, N.K., Newcomb, R.W.: Tellegon’s theorem and multivariable realizability theory. Int. J. Electron. 36 (1974), 417-425.
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.
Hsu, J.C., Meyer, A.U.: Mordern Control principles and Applications. McGrawHill, New York, 1968.
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.
Ding, W., Qi, L., Wei, Y.: M-tensors and nonsingular M-tensors. Linear Algebra Appl. 439 (2013), 3264-3278.
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.