Further Results for Z $Z$ -Eigenvalue Localization Theorem for Higher-Order Tensors and Their Applications
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Further Results for Z-Eigenvalue Localization Theorem for Higher-Order Tensors and Their Applications Liang Xiong1 · Jianzhou Liu1
Received: 13 December 2019 / Accepted: 2 May 2020 © Springer Nature B.V. 2020
Abstract In this paper, we present some new Z-eigenvalue inclusion theorem for tensors by categorizing the entries of tensors, and prove that these sets are more precise than existing results. On this basis, some lower and upper bounds for the Z-spectral radius of weakly symmetric nonnegative tensors are proposed, which improves some of the existing results. As applications, we give some estimates of the best rank-one approximation rate in weakly symmetric nonnegative tensors and the maximal orthogonal rank of real orthogonal tensors, and our results are more precise than existing result in some situations. In particular, for a given symmetric multipartite pure state with nonnegative amplitudes in real field, some theoretical lower and upper bounds for the geometric measure of entanglement are also derived in terms of the bounds for Z-spectral radius. Numerical examples are given to illustrate validity and superiority of our results. Keywords Z-eigenvalue · Nonnegative tensors · Inclusion set · Spectral radius · Weakly symmetric · Best rank-one approximation rate · Geometric measure of quantum entanglement Mathematics Subject Classification 15A18 · 15A69 · 15A21
1 Introduction Let C(R) be he set of all complex (real) numbers, and denote the set N = {1, 2, . . . , n} for a positive integer n ≥ 2. A complex (real) tensor A of order m dimension n is a multidimensional array consisting of nm complex (real) entries ai1 i2 ···im ∈ C(R), where ij = 1, 2, . . . , n for j = 1, 2, . . . , m. Obviously, a tensor of order 1 is a vector, and a
B J. Liu
[email protected] L. Xiong [email protected]
1
School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan 411105, China
L. Xiong, J. Liu
tensor of order 2 is a matrix. For a tensor A = (ai1 i2 ···im ) ∈ R[m×n] , A is nonnegative (positive) if every its entry ai1 i2 ···im ≥ (>)0, and A is called symmetric [1] if its element ai1 i2 ···im is invariant under any permutation of m indices (i1 , i2 , . . . , im ). A tenor A = (ai1 i2 ···im ) ∈ R[m×n] is weakly symmetric [2] if the associated homogeneous polynomial n
Ax m =
ai1 i2 ···im xi1 xi2 · · · xim
i1 ,i2 ,...,im =1
satisfied ∇ Ax m = mAx m−1 , where x = (x1 , x2 , . . . , xn )T ∈ Rn , and Ax m−1 is an n dimension vector in Cn , whose i-th component is
Ax m−1
i
=
n
aii2 ···im xi2 · · · xim .
i2 ,i3 ,...,im =1
It’s worth noting that a symmetric tensor must be a weakly symmetric tensor, but not vice versa. Therefore, some conclusions that are valid for symmetric tensors will no longer be applicable for weakly symmetric tensors. In particular, a tensor A of order m dimension n is said to be a rank-one tensor, if there exists a number of vectors such that it can be expressed as an outer produc
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