Permutation transformations of tensors with an application

Li et al. SpringerPlus (2016) 5:2023 DOI 10.1186/s40064-016-3720-1 Open Access RESEARCH Permutation transformations of tensors with an application Yao‑Tang Li*†, Zheng‑Bo Li†, Qi‑Long Liu† and Qiong Liu† *Correspondence: † Yao-Tang Li, Zheng-Bo Li, Qi-Long Liu and Qiong Liu contributed equally to this work School of Mathematics and Statistics, Yunnan University, Kunming 650091, People’s Republic of China Abstract The permutation transformation of tensors is introduced and its basic properties are discussed. The invariance under permutation transformations is studied for some important structure tensors such as symmetric tensors, positive definite (positive semidefinite) tensors, Z-tensors, M-tensors, Hankel tensors, P-tensors, B-tensors and H-tensors. Finally, as an application of permutation transformations of tensors, the canonical form theorem of tensors is given. The theorem shows that some problems of higher dimension tensors can be translated into the corresponding problems of lower dimension weakly irreducible tensors so as to handle easily. Keywords: Permutation transformation, Structure tensor, Weakly irreducible tensor, Canonical form Mathematics Subject Classification: 15A69, 12E05, 12E10 Background The study of tensors with their various applications has attracted extensive attention and interest, since the work of Qi (2005) and Lim (2005). Lately, the research topic on structure tensors has also attracted much attention, such as symmetric tensors (Qi 2005), P(P0 )-tensors (Song and Qi 2014), B(B0 )-tensors (Song and Qi 2014), Z-tensors (Zhang et al. 2014), (strong) M-tensors (Zhang et al. 2014), H-tensors (Li et al. 2014) and so on. In the researches on tensors with its application, the reducibility and higher dimension of tensors are two important factors to cause difficulties. Therefore, it is interesting that how to translate problems of higher dimension reducible tensors into the corresponding problems of lower dimension irreducible tensors. As we all know, the permutation transformation of matrices plays a very important role in linear algebra and matrix theory. Some problems of higher dimension reducible matrices can be translated into the corresponding problems of lower dimension irreducible matrices by using the permutation transformation of matrices. Inspired by this, we introduce permutation transformations of tensors, and discuss its basic properties and and their applications in this paper. In the next section, we will introduce the permutation transformation of tensors and give its expression. In third section, we will discuss basic properties of permutation transformations of tensors. In fourth section, we will discuss the invariance under © The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Li et al. SpringerPlus (2016) 5:2023 Page 2 of 15 permutation transformations for some important structure tensors such as symmetric tensors, positive definite tensors, M-tensors, Hankel tensors, P-tensors, B-tensors, H-tensors and so on. In fifth section, we will give the canonical form theorem of tensors and a numerical example which shows that some problems of higher dimension tensors can be translated into the corresponding problems of lower dimension weakly irreducible tensors by using permutation transformations. Finally, we draw some conclusions in the last section. Permutation transformations of tensors and its expression For a positive integer n, let [n] = {1, 2, . . . , n}. An order m tensor A = (ai1 ...im ) ∈ Cn1 ×n2 ×···×nm is a multidimensional array with n1 n2 . . . nm entries, where ij ∈ [nj ], j ∈ [m]. Especially, an order m dimension n tensor A = (ai1 ...im ) over the complex field C (real field R) consists of nm complex (real) entries: ai1 ...im ∈ C (R), where ij ∈ [n] for j ∈ [m] (Chang et al. 2008; De Lathauwer et al. 2000; Liu et al. 2010; Ng et al. 2009; Zhang and Golub 2001). It is obvious that a matrix is an order 2 tensor. We shall denote the set of all complex (real) order m dimension n tensors by C[m,n] (R[m,n], respectively). Definition 1 Let A = (ai1 ...im ) ∈ C[m,n] , B = (bi1 ...im ) ∈ C[m,n], and k ∈ C. Define (i) A + B = (ai1 ...im + bi1 ...im ). (ii) k A = (kai1 ...im ). Remark 1 Obviously, both C[m,n] and R[m,n] are linear spaces about the addition and the multiplication in Definition 1. Definition 2 (Qi 2005) A tensor A = (ai1 ...im ) ∈ R[m,n] is called a symmetric tensor if its entries ai1 ...im are invariant under any permutation of their indices. Denote the set of all real order m dimension n symmetric tensors by S[m,n]. Furthermore, S[m,n] is a linear subspace of R[m,n]. An order m dimension n tensor is called the unit tensor (Yang and Yang 2010), denoted by I , if its entries are δi1 ...im for i1 , . . . , im ∈ [n], where  1, if i1 = · · · = im , δi1 ...im = 0, otherwise. Let A = (ai1 ...im ) ∈ R[m,n] and x ∈ Rn. Then Axm is a homogeneous polynomial of degree m, defined by  Ax m = ai1 ...im xi1 . . . xim . i1 ,...,im ∈[n] A tensor A ∈ R[m,n] is called positive semidefinite (Song and Qi 2014) if for any vector x ∈ Rn , Axm ≥ 0, and it is called positive definite if for any nonzero vector x ∈ Rn , Axm > 0. Li et al. SpringerPlus (2016) 5:2023 Page 3 of 15 Now, we give the definition of permutation transformation of tensors. Definition 3 Let A = (ai1 ...im ) ∈ C[m,n] and π be a permutation on [n], we define Pπ : C[m,n] → C[m,n] by Pπ (A) = (aπ(i1 )...π(im ) ). Pπ is called as a permutation transformation on C[m,n], and is simply called as a permutation transformation. Pπ (A) is called as the image of A under Pπ. Remark 2 Pπ is called as a permutation transformation on R[m,n] if C[m,n] is replaced by R[m,n] in Definition 3. Definition 4 Let A = (ai1 ...im ) ∈ C[m,n] and π −1 be the inverse permutation of π on [n], we define Pπ−1 : C[m,n] → C[m,n] by Pπ−1 (A) = Pπ −1 (A). Pπ−1 is called as the inverse permutation transformation of Pπ on C[m,n], and is simply called as the inverse permutation transformation. For further discussing property of the permutation transformation of tensors, we introduce the following general product of two n-dimensional tensors defined in Shao (2013). For the sake of simplicity, we sometime use the following “condensed notation” for the subscripts of the tensor. For example, we will write ai1 i2 ...im as ai1 α, where α = i2 . . . im ∈ [n]m−1and [n]m−1 is m − 1 dimensional array whose every element varies from 1 to n. Definition 5 (Shao 2013) Let A and B be order m ≥ 2 and order k ≥ 1, dimension n tensors, respectively. Define the product A · B (sometimes simplified as AB) to be the following tensor C of order (m − 1 (...truncated)