Abstract
We explore the singular value decomposition and full rank decomposition of arbitrary-order tensors using reshape operation. Here the treatment of arbitrary-order is a straightforward task due to the nature of the isomorphism between arbitrary-order tensor spaces and matrix spaces. Within the framework, the Moore-Penrose and weighted Moore-Penrose inverse for arbitrary-order tensors via the Einstein product is defined. Further, we elaborate on this theory by producing a few identities involving the weighted Moore-Penrose inverses of tensors. We then obtain a few necessary and sufficient conditions of the reverse-order law for weighted Moore-Penrose inverses of arbitrary-order tensors.

1. Introduction
Background and motivation
Tensors or hypermatrix are multidimensional generalizations of vectors and matrices, and have attracted tremendous interest in the recent years [16, 22, 26, 25, 27] . Indeed, multilinear systems are closely related with tensors and such systems are encountered in a number of fields of practical interests, i.e., signal processing [20, 29, 10] , scientific computing [4, 28, 5] , data mining [9] , compression and retrieval of large structured data [11, 6] . Further, the Moore-Penrose inverse of tensors plays an important role for solving such multilinear systems [2, 15, 31] and the reverse-order law for the Moore-Penrose inverses of tensors yields a class of interesting problems that are fundamental in the theory of generalized inverses of tensors [24] . In view of this, multilinear algebra is gaining more and more interest [15, 1, 22, 18, 20] , specifically, the recent works (e.g., [2, 5, 14, 24, 30, 31] ), motivate us to study in the framework of arbitrary-order tensors.
Background and motivation
Let C I 1 ×···×I N (R I 1 ×···×I N ) be the set of order N and dimension I 1 × · · · × I N tensors over the complex (real) field C(R). Let A ∈ C I 1 ×···×I N is a multiway array with N-th order tensor, and I 1 , I 2 , · · · , I N are dimensions of the first, second,· · · , Nth way, respectively. Indeed, a matrix is a second order tensor, and a vector is a first order tensor. We refer to R m×n as the set of all real m × n matrices. Note that throughout the paper, tensors are represented in calligraphic letters like A, and the notation (A) 
Background and motivation
The Einstein product is not commutative but associative, and distributes with tensor addition. Further, cancellation is not worked but there is a identity tensor I for multiplication. This product is used in the study of the theory of relativity [12] and in the area of continuum mechanics [19] . On the other hand, one of the most successful developments in the world of linear algebra is the concept of the Singular Value Decomposition (SVD) of matrices [3] . This concept gives us important information about a matrix such as its rank, an orthonormal basis for the column or row space, and reduction to diagonal form [32] . Since tensors are natural multidimensional generalizations of matrices and lot of applications involving arbitraryorder tensors. Further, the problem of decomposing tensors is approached in a variety of ways by extending the SVD, and extensive studies (e.g., [8, 16, 18, 20, 29, 21] ) have exposed many aspects of these decomposition and its applications. However, the existing framework of SVD of tensor appears to be insufficient and/or inappropriate. The aim of this paper is to present a proper generalization of the SVD of arbitrary-order tensors under Einstein tensor product. Although the existing form [5] of the SVD is well suited for square tensor, as follows. 
Background and motivation
2 where T I,J,I,J (R) = {A ∈ R I×J×I×J : det(f (A)) = 0}. In general for any even order tensor, the transformation is defined as; f :
Background and motivation
Using the above Definition and Theorem 3.17 in [5] gives the SVD of a tensor A ∈ R I×J×I×J , which can be extended only to any square tensor, i.e., for A ∈ R
Background and motivation
Extension of the SVD for an arbitrary-order tensor using this method [5] is impossible, since f is not homomorphism for even-order and/or arbitrary-order tensors. Conspicuously, the Einstein product is not defined for the following two even-order tensors, A ∈ R
Background and motivation
and B ∈ R I 1 ×I 2 ×J 1 ×J 2 , i.e., A * 2 B is not defined. Similar predicament occurs in [21] for arbitrary-order tensors. In order to get the homomorphism (defined in Lemma 3.1 part (2) in [5] and map φ defined in Section 2.2 in [21] ) the tensor should be a square tensor, i.e., A ∈ R I 1 ×I 2 ×···×I N ×I 1 ×J 2 ×···×I N . Therefore, our aim is to find the SVD for any arbitrary order tensors using reshape operation, which is discussed in next section. In addition to that, recently there has been increasing interest in analyzing inverses and generalized inverses of tensors based on different tensor products [15, 5, 31] . The representations and properties of the ordinary tensor inverse were introduced in [5] . This interpretation is extended to the Moore-Penrose inverse of tensors in [31] and investigated a few characterizations of different generalized inverses of tensors via Einstein product in [2] . At the last, Behera and Mishra [2] posed the open question: "Does there exist a full rank decomposition of tensors ? If so, can this be used to compute the Moore-Penrose inverse of a tensor". Liang and Zheng [21] investigated this question, however, it is not suitable for arbitrary-order tensor. Therefore, it is appropriate to employ an isomorphism operator, to get a proper full rank decomposition of an arbitrary-order tensor. The reshape operation provide a natural tool to deal with such problems, which is discussed in next section.
Background and motivation
Recently, Panigrahy and Mishra [23] investigated the Moore-Penrose inverse of a product of two tensors via Einstein product. Using such theory, Stanimirovic, at. al [30] introduced some basic properties of the range and null space of multidimensional arrays, and the effective definition of the tensor rank, termed as reshaping rank. In this respect, Panigrahy et al. [24] obtained a few necessary and sufficient conditions for the reverse order law for the Moore-Penrose inverses of tensors, which can be used to simplify various tensor expressions that involve inverses of tensor products. Corresponding representations of the weighted Moore-Penrose inverse [14] of an even-order tensor was introduced at this juncture.
Background and motivation
In this paper, we study weighted Moore-Penrose inverse of an arbitrary-order tensor. This study can lead to the enhancement of the computation of SVD and full rank decomposition of arbitrary-order tensor using reshape operator. In this regard, we discuss some identities involving the weighted Moore-Penrose inverses of tensors and then obtain a few necessary and sufficient conditions of the reverse order law for the weighted Moore-Penrose inverses of arbitrary-order tensors via the Einstein product.
Outline
We organize the paper as follows. In the next subsection, we introduce some notations and definitions which are helpful in proving the main results. Section 3 discusses the main results and has three parts. In this section, we introduce SVD and full rank decomposition of an arbitrary-order tensor using reshape operation. Within the framework, the MoorePenrose and generalized weighted Moore-Penrose inverse for arbitrary-order tensor is defined. Further, in this section, we obtain several identities involving the weighted Moore-Penrose inverses of tensors via Einstein product. The Section 3 contains a few necessary and sufficient conditions of the reverse-order law for the weighted Moore-Penrose inverses of tensors.
Notations and definitions
For convenience, we first briefly explain a few essential facts about the Einstein product of tensors, which are found in [2, 5, 31] . For a tensor A = (a i 1 ..
Notations and definitions
the transpose of A, denoted by A T . The definition of a diagonal tensor was introduced as a square tensor in [5] (Definition 3.12) and as an even-order tensor in [31] . Here we introduce arbitrary-order diagonal tensor, as follows.
Notations and definitions
Using this notation we recall the definition of an identity tensor below.
Definition 1.2. (Definition 3.13, [5]) A tensor
is called a unit tensor or identity tensor.
Definition 1.2. (Definition 3.13, [5]) A tensor
Note that throughout the paper, we denote I M ,I L and I R as identity tensors in the space
Definition 1.2. (Definition 3.13, [5]) A tensor
tensor O denotes the zero tensor if all the entries are zero. A tensor A ∈ C Next we present the definition of the reshape operation, which was introduced earlier in [30] . This is a more general way of rearranging the entries in a tensor (it is also a standard Matlab function), as follows.
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
where
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
reshaping is the mapping defined as:
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
where the matrix A ∈ C I 1 ···I M ×J 1 ···J N and the tensor A ∈ C
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
Further Lemma 3.2 in [30] defined the rank of a tensor, A, denoted by rshrank(A) as
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
Continuing this research, Stanimirovic et. al. in [30] discussed the homomorphism properties of the rsh function, as follows,
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
where
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
An immediate consequence of the above Lemma is follows as:
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
Existence of SVD of any square tensor is discussed in [5] , in process of this framework, Jun and Wei [14] defined Hermitian positive definite tensors, as follows.
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
5
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
where D ∈ C I 1 ×···×I N ×I 1 ×···×I N is a diagonal tensor with positive diagonal entries, then P is said to be Hermitian positive definite tensor.
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
Further, Jun and Wei [14] defined the square root of a Hermitian positive definite tensor, P, as follows.
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
where D 1/2 is the diagonal tensor, which obtained from D by taking the square root of all its diagonal entries. Notice that P 1/2 is always non-singular and its inverse is denoted by
Definition 1.2. (Definition 3.1, [30]): The 1-1 and onto reshape map, rsh, is defined as:
we now recall the definition of the range and the null space of arbitrary order tensors.
Definition 1.4. ( Definition 2.1,[30]): The null space and the range space of
are defined as follows: 
Main results
Mathematical modelling of problems in science and engineering typically involves solving multilinear system. This becomes particularly challenging for problems having an arbitraryorder tensor. However, until now the Moore-Penrose inverses have been limited to even-order tensors. It is thus of interest to study the theory of Moore-Penrose inverse of an arbitraryorder tensor via the Einstein product.
Moore-Penrose inverses
One of the most widely used methods is the SVD to compute Moore-Penrose inverse. Here we present a proper generalization of the SVD via the Einstein product.
Moore-Penrose inverses
where
0, otherwise
where
0, otherwise
In the context of the SVD of the matrix A, one can write A = UDV * , where
0, otherwise
From relations (7) and (8), we can write
0, otherwise
where 
0, otherwise
the following four tensor equations: By straightforward derivation, the following results can be obtained, which also hold (Lemma 2.3 and Lemma 2.6) in [2] for even-order tensor.
Weighted Moore-Penrose inverse
Weighted Moore-Penrose inverse of even-order tensor, A ∈ C I 1 ×···×I K ×J 1 ×···×J K was introduced in [14] , very recently. Here we have discussed weighted Moore-Penrose inverse for an arbitrary-order tensor via Einstein product, which is a special case of generalized weighted Moore-Penrose inverse.
and a pair of invertible and Hermitian tensors
said to be the generalized weighted Moore-Penrose inverse of A with respect to M and N , if Y satisfies the following four tensor equations However, the generalized weighted Moore-Penrose inverse Y does not always exist for any tensor A, as shown below with an example. 
Example 2.3. Consider tensors
Further, since (A T * 2 A)
Example 2.3. Consider tensors
this follows that A T * 2 M * 2 A is invertiable, which is a contradiction.
Example 2.3. Consider tensors
At this point one may be interested to know when does the generalized weighted MoorePenrose inverse exist ? The answer to this question is explained in the following theorem. 
Penrose inverse of an arbitrary-order tensor A exists and is unique, i.e., there exist a unique tensor
where M (15), then the following identities are hold.
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
where A is any arbitrary-order tensor. Now we define weighted conjugate transpose of a arbitrary-order tensor, as follows.
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
positive definite tensors, the weighted conjugate transpose of A ∈ C
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
Next we present the properties of the weighted conjugate transpose of any arbitrary-order tensor, A ∈ C I 1 ×···×I M ×J 1 ×···×J N , as follows.
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
Adopting the result of Lemma 2.9(b) and the definition of the weighted Moore-Penrose inverse, we can write the following identities.
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
Using the Lemma 3.17 in [24] on two invertibale tensors B ∈ C I 1 ×···×I M ×I 1 ×···×I M and C ∈ C J 1 ×···×J N ×J 1 ×···×J N , one can write the following identities
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
where A is the arbitrary-order tensor, i.e., A ∈ C I 1 ×···×I M ×J 1 ×···×J N . Further, using Eq. (16) and Corollary 2.5, we get following results.
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
In the following Theorem, a lot of well-known and important facts and properties concerning the weighted Moore-Penrose inverse are collected.
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
Further, using Lemma 2.7(b) and Lemma 2.6(b) along with the fact (
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
we obtain,
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
Replacing U and V by M 1/2 * M U and V * L N −1/2 respectively on the above result, we get
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
Substituting the above result in Eq. (15) we get the desired result.
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
When investigating on the reverse-order law, we find some interesting mixed-type reverse order laws.
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
we get,
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
Substituting the above result in Eq. (15) one can get the desired result.
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
Hermitian positive definite tensors, Then
Using the Definition 2.2 and following the Lemma 2 in [14] one can write (A
Further, using Lemma 2.11[(a),(b)] and Eq. (15), and Eq. (19) we have 
The full rank decomposition
The tensors and their decompositions originally appeared in 1927, [13] . The idea of decomposing a tensor as a product of tensors with a more desirable structure may well be one of the most important developments in numerical analysis such as the implementation of numerically efficient algorithms and the solution of multilinear systems [16, 7, 22, 17] . As part of this section, we focus on the full rank decomposition of a tensor. Unfortunately, It is very difficult to compute tensor rank. But the authors of in [30] introduced an useful and effective definition of the tensor rank, termed as reshaping rank. With the help of reshaping rank, We present one of our important results, full rank decomposition of an arbitrary-order tensor.
The full rank decomposition
where rshrank(F ) = rshrank(G) = rshrank(A) = r = H 1 · · · H R . This is called the full rank decomposition of the tensor A.
Proof. Let the matrix
Then we have, rank(A) = r. Suppose that the matrix A has a full rank decompositions, as follows,
Proof. Let the matrix
where F ∈ C I 1 ···I M ×H 1 ···H R is a full column rank matrix and G ∈ C H 1 ···H R ×J 1 ···J N is a full row rank matrix. From Eq. (8) and Eq. (21) we obtain
Proof. Let the matrix
where
Proof. Let the matrix
where 
Proof. Let the matrix
Now, the following theorem expressed the weighted Moore-Penrose inverse of a tensor A ∈ C I 1 ×···×I M ×J 1 ×···×J N in form of the ordinary tensor inverse.
Theorem 2.12. If the full rank decomposition of a tensor
by Eq. (20) , then the weighted Moore-Penrose inverse of A can be written as Here B is in the form of full rank decomposition, as both M 1/2 and N 1/2 are invertible. Now, from Lemma 2.13, we get
Theorem 2.12. If the full rank decomposition of a tensor
In particular when the arbitrary-order tensor, A is either left invertible or right invertible, we have the following results. Moreover,
Theorem 2.12. If the full rank decomposition of a tensor
Proof. Suppose the tensor, A ∈ C I 1 ×···×I M ×J 1 ×···×J N has two full rank decompositions, as follows,
Theorem 2.12. If the full rank decomposition of a tensor
where F , F 1 ∈ C I 1 ×···×I M ×H 1 ×···×H R and G,
Theorem 2.12. If the full rank decomposition of a tensor
is equivalent to C = B −1 . Therefore
Theorem 2.12. If the full rank decomposition of a tensor
Similarly G † 1 = G † * R B.
Reverse order law
In this section, we discuss some necessary and sufficient condition for reverse order law of weighted Moore-Penrose inverse. 

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