1 Introduction and motivation
Let be the set of order and dimension tensors over the complex field and the entry of tensor is denoted by . Here is a multiway array with -th order tensor, and are dimensions of the first, second, , th way, respectively. Note that throughout the paper, tensors are represented in calligraphic letters like , and the notation represents the scalars. The Einstein product (ein ) of tensors and is defined by the operation via
The Einstein products ideally suited to addressing the problem of finding multilinear structure in multiway data-sets. Such multiscale issues are encountered in many fields of practical interest RD ; ein ; lai ; sun .
In connection with tensors and multilinear systems, Brazell et al. BraliNT13 introduced the notion of the ordinary tensor inverse, as follows. A tensor is called the inverse of if it satisfies . It is denoted by . As a consequence, most of the current efforts are focused on developing different type of inverses of tensors depending on the application BraliNT13 ; ishteva2011best ; ShiWeL13 . Sun et al. sun introduced Moore-Penrose inverse of tensors and discussed the existence and uniqueness of the inverse with the Einstein product. In this context, Behera and Mishra RD discussed -inverses (for ) of even-order tensors and general solutions of multilinear systems. Subsequently, the LU and the Schur decompositions in the tensor were discussed in liang2019 . Further, solutions of a multilinear system represented by generalized inverses of tensors were discussed in sun2018 . Using such theory of the Einstein product, Ji and Wei weit2 introduced weighted Moore-Penrose inverse of tensors and presented a few characterization of the least-squares solutions to a multilinear system. Further, full rank decomposition liangBing2019 of arbitrary-order tensors using reshape operation is discussed in Behera2018 . At the beginning of this work, Stanimirovic at el. stan discussed the effective definition of the tensor rank and outer inverses of tensors. In addition to this, the Drazin inverse of even-order tensors and its applications via the Einstein product is investigated in RAJ ; Ji18 . Recently, Liang and Zheng liang2018 defined an iterative algorithm for solving Sylvester tensor equation using such theory.
On the other hand, the concept of tensor inversion technique has opened new perspectives for solving multilinear systems across many fields in science and engineering SilL08 ; kolda ; Kru77 ; LalmV00 ; liang2018 ; SidLaP17 ). The inversion of tensor impinges upon a definition of tensor-tensor multiplication, and the treatment of multilinear system is a straightforward task due to the nature of the Einstein product. However, this becomes particularly challenging for problems having singular or arbitrary-order tensor. Generalized inverses of tensors take advantage to do this RD ; RAJ ; BraliNT13 ; Ji18 . In this context, Sahoo at al. JRPKH introduced the core inverses of tensors and discussed some characterizations of inverses. At last, they posed the open question regarding reverse-order law for core inverse of the tensors. It will be more interesting if we study the reverse-order law for the core inverse of tensors.
It is well known that where and are a pair of invertible tensors such that their Einstein product is invertible. This equality is called the reverse-order law for the ordinary inverse. In general, this equality doesn’t hold if we replace the ordinary inverse by the generalized inverse of tensors. This reverse-order law is a class of interesting problems that are fundamental in the theory of generalized inverses of tensors. In this context, Panigrahy at al RDP first gave a necessary and sufficient condition of the reverse-order law for the Moore-Penrose inverse of tensors via the Einstein product. Since then, many authors studied the reverse-order law for various classes of generalized inverses of tensors Mispa18 ; RDP ; panigrahy2019 .
Indeed, the reverse-order law, first studied a necessary and sufficient condition for the Moore-Penrose inverse in from of rectangular matrices in greville1966 . Baskett and Katz baskett1969 then discussed the same theory for matrices, where of rank is called , if and , the conjugate transpose of , have the same null spaces. Further, Deng deng2011 investigated some necessary and sufficient conditions of the reverse-order law for the group inverse of linear bounded operators on Hilbert spaces. The reverse-order law was also studied for other generalized inverses of matrices (Barwick1974 , Cao2004 , sun1998 ). Under the influence of the vast work on the reverse-order law and solution of multilinear systems. Here we discuss reverse-order law for core inverse of tensors and a few characterizations of the multilinear systems. The results are new even in the case of matrices.
We organize the paper as follows. In the next section, we discuss some notations and definitions along with a few preliminary results, which are helpful in proving the main results. In Section 3 contains a few necessary and sufficient conditions of the reverse-order law for the core inverses of tensors via the Einstein product. Further, in section 4, we prove several results concerning mixed-type of the reverse-order law. General solutions of multilinear systems using core inverse of the tensors are addressed in section 5, and the conclusion is outlined in section 6.
2.1 Definitions and terminology
For convenience, we first briefly explain some additional notations to increase the efficiency of the presentation, as follows.
In view of the above notation, the tensor will be shortly denoted by Further, the following notation is useful:
Accordingly, the tensor will be denoted in a simpler form as . For a tensor the transpose of is denoted by and defined as . Further, we denote by zero tensor and is the identity tensor. A tensor is Hermitian if skew-Hermitian if , and orthogonal if . Let us recall the definition of the Moore-Penrose inverse of a tensor which was introduced in sun , as follows.
( Definition 2.2, sun ) Let . The tensor satisfying
(1) (3) (4) ,
is called Moore-Penrose inverse of and denoted by .
Further, the authors of sun also discussed the existence and uniqueness of the inverse. Note that, the Moore-Penrose inverse always exists and unique.
Now, we recall the index of a tensor which was first discusses in Ji18 . The index of a tensor is defined as the smallest positive integer such that and it is denoted by ind. If then the tensor is called index one or group, or core tensor. Let the tensor and ind, then the Drazin inverse of is defined in Ji18 as follows.
Let and ind. A tensor satisfying
is called Drazin inverse of .
The Drazin inverse is uniques if exists and denoted by . In particular, when , the Drzain inverse is called group inverse of and denoted by . Subsequently, the notion of core inverse for complex tensors was introduced by Sahoo et. al. JRPKH , recently, as follows.
Let . A tensor satisfying
(3) (6) (7) ,
is called core inverse of .
The core inverse is unique whenever it exists and denoted by . If a tensor is core tensors then its core inverse always exist and we call the tensor , is core invertible. Further, a tensor is called EP sahoo2019 if .
Let . If a tensor satisfies equation (i), where , then is called -inverse of . The set of -inverse of the tensor is denoted by . For example, , and respectively for set of generalized inverse, generalized reflexive inverse of , and inverse of We use the notation , is a fixed element of . For example, , respectively one element of and . The range and null space of a tensor was introduced in Ji18 ; stan , and defined as
We conclude this subsection with the definition of Kronecker product and one result using the product, which are essential to prove our main results.
The Kronecker product of
and , denoted by
is a ‘Kr-block tensor’ whose
subblock is obtained via multiplying all the entries of by
a constant where
2.2 Prerequisite results
Now, we discuss some preliminary results, which will be used in our subsequent sections. Out of which, few known results directly collected from literature. We first recall the result based on the Drazin inverse, as follows.
Lemma 2.6 (Theorem 3.13, Raj ).
Let , and ind. Then .
Now we present a few impressive results from JRPKH in which the core inverse of a tensor in linked with other types of inverses, i.e., Moore-Penrose inverse, group inverse, and EP.
Lemma 2.7 (Theorem 3.3, Jrpkh ).
Let be a core tensor. If there exists a tensor such that , then .
Lemma 2.8 (Theorem 3.2, 3.5, Jrpkh ).
Let be a core tensor. Then the following statements are equivalent:
Further, the authors of stan , proved the following result for the range space.
Lemma 2.9 (Lemma 2.2, stan ).
Let , Then if and only if there exists such that .
Using Lemma 2.9, we can easily show the following result.
Let . Then exists if and only if for some , .
Let and ind. Then for some , .
Since exists, by Lemma 2.10, we have and . To claim the result, it is enough to show is the group inverse of . Now
Hence is the group inverse of . ∎
If is -inverse of , then . Conversely, if for some , then we can easily get
. Therefore, we obtain the following result.
Let and . Then is -inverse of if and only if .
Now we recall a characterization of the core-tensor, as follows.
Lemma 2.13 (Proposition 3.1 and Corollary 3.7, Jrpkh ).
Let be a core tensor. If a tensor satisfying any one of the condition
, , and ,
, , and ,
then is the core inverse of .
Next, we discuss a few results based on commutative of two tensors.
Lemma 2.14 (Theorem 3.11 (a) when , Raj ).
Let and . If , then and . Further, .
Let . If and , then .
Let and ind. If there exist a tensor such that and , then .
3 Reverse-order law
In this section, we discuss various necessary and sufficient conditions of the reverse-order law for the core inverses of tensors. The very first result obtained below deals with sufficient condition for the reverse-order law of tensors
Let be core tensors. If and , then
If we consider in the above theorem we get the following result.
Let be a core tensors. Then and .
Let be a core tensor. If , then .
Note that, if and , then from Corollary 2.16, we can easily get and . In conclusion, we state the following as a corollary.
Let be core tensors. If and , then
Converse of the above result is not true, and is shown in the following example.
Let and and be two core tensors such that
respectively. Then and , where