Z $Z$ -Eigenvalue Localization Sets for Even Order Tensors and Their Applications
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Z-Eigenvalue Localization Sets for Even Order Tensors and Their Applications Caili Sang1,2 · Zhen Chen1
Received: 27 August 2019 / Accepted: 6 November 2019 © Springer Nature B.V. 2019
Abstract Firstly, a new Geršgorin-type Z-eigenvalue localization set with parameters for even order tensors is presented. As an application, some sufficient conditions for the positive (semi-)definiteness of even order real symmetric tensors are obtained. Secondly, by selecting appropriate parameters an optimal set is obtained and proved to be tighter than some existing results. Thirdly, as another application, new upper bounds for the Z-spectral radius of even order weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results. Keywords Nonnegative tensors · Z-eigenvalues · Z-spectral radius · Localization sets · Positive definiteness Mathematics Subject Classification (2010) 15A18 · 15A42 · 15A69
1 Introduction Let m and n be two positive integers, m, n ≥ 2, [n] be the set {1, 2, . . . , n}, C (or, respectively, R) be the set of all complex (or, respectively, real) numbers, R[m,n] be the set of all order m dimension n real tensors. Let A = (ai1 i2 ···im ) ∈ R[m,n] , that is, ai1 i2 ···im ∈ R,
ij ∈ [n], j ∈ [m].
A is called symmetric [1] if ai1 ···im = aiπ(1) ···iπ(m) , ∀π ∈ Πm , where Πm is the permutation group of m indices.
B Z. Chen
[email protected] C. Sang [email protected]; [email protected]
1
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, P.R. China
2
College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, P.R. China
C. Sang, Z. Chen
If there exists λ ∈ C and x = (x1 , x2 , . . . , xn )T ∈ Cn \{0} such that A x m−1 = λx
and
x T x = 1,
(1)
where A x m−1 is an n dimension vector whose ith component is A x m−1 i = aii2 ···im xi2 · · · xim , i2 ,...,im ∈[n]
then λ is called an E-eigenvalue of A and x an E-eigenvector of A associated with λ. If λ and x are all real, then λ is called a Z-eigenvalue of A and x a Z-eigenvector of A associated with λ; for details, see [1–3]. Let σ (A ) (or, respectively, σE (A )) be the set including all Z-eigenvalues (or, respectively, E-eigenvalues) of A . Tensor A defines an mth-degree homogeneous polynomial f (x) := A x m = ai1 i2 ···im xi1 xi2 · · · xim . i1 ,i2 ,...,im ∈[n]
We say that an even-order real symmetric tensor A is positive (semi-)definite if f (x) is positive (semi-)definite. When m is even, the positive definiteness of such a homogeneous polynomial form f (x) plays an important role in the stability study of nonlinear autonomous systems via Lyapunov’s direct method in automatic control [4–8]. Moreover, it can be observed from Theorem 8.5 in [3] that if one wants to judge that whether the strong ellipticity condition of anisotropic elastic materials holds or not, he needs to judge the positive definiteness of three second order symmetric tensors, a fourth-order symmetric tensor and a sixth-order symmetric tens
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