Further results on the Drazin inverse of even-order tensors

Further results on the Drazin inverse of even-order tensors

Ratikanta Behera , Ashish Kumar Nandi , Jajati Keshari Sahoo Department of Mathematics and Statistics,
Indian Institute of Science Education and Research Kolkata,
E-mail: ratikanta@iiserkol.ac.in Department of Mathematics,
BITS Pilani, K.K. Birla Goa Campus, Goa, India
E-mail: ashish.nandi123@gmail.com
E-mail: jksahoo@goa.bits-pilani.ac.in
Abstract

The notion of the Drazin inverse of an even-order tensor with the Einstein product was introduced, very recently [J. Ji and Y. Wei. Comput. Math. Appl., 75(9), (2018), pp. 3402-3413]. In this article, we further elaborate this theory by producing a few characterizations of the Drazin inverse and the W-weighted Drazin inverse of tensors. In addition to these, we compute the Drazin inverse of tensors using different types of generalized inverses and full rank decomposition of tensors. We also address the solution to the multilinear systems using the Drazin inverse and iterative (higher order Gauss-Seidel) method of tensors. Besides this, the convergence analysis of the iterative technique is also investigated within the framework of the Einstein product.

keywords:
Einstein product, Tensor inversion, Drazin inverse, W-weighted Drazin inverse, Multilinear system.
AMS Subject Classifications: 15A69; 15A09
journal: Comput. Math. Appl.

1 Introduction

The Drazin inverse plays an important role in various applications in singular differential liang2019 () and difference equations meyer (), Markov chains camp (); meyer (), investigation of Cesaro-Neumann iterations hart82 (), Cryptography levin (), and iterative methods miller (); wei98 (). Specifically, the Drazin inverse extensively used to solve the system of linear equations, where the iterative schemes lead from matrix splitting. However, many interesting physical systems are required to store huge volumes of multidimensional data, and in recognition of potential modeling accuracy, matrix representation of data analysis is not enough to represent all the information. Tensors are natural multidimensional generalizations of matrices, which efficiently solve these problems. In this context, Ji and Wei Wei18 () introduced the Drazin inverse of an even-order tensor through the core-nilpotent decomposition to solve singular tensor equations. It will be more applicable if we study the characterization of the Drazin inverse of tensors, and hence this inverse of tensors will open different paths in the above areas.

On the other hand, the concept of tensor-structured numerical methods have opened new perspectives for solving multilinear systems, recently. Many computational and theoretical problems require different types of generalized inverses when a tensor is singular or arbitrary order. The authors of bral () discussed the representations and properties of the ordinary tensor inverse and introduced tensor-based iterative methods to solve high-dimensional Poisson problems in the multilinear system framework. This interpretation is extended to the Moore-Penrose inverse of tensors in bm (); liang2019 () and discussed the solution of multilinear systems and tensor nearness problem associated with tensor equations. Using such theory of Einstein product, Liang et al. liang () investigated necessary and sufficient conditions for the invertibility of tensors, and proposed the LU and the Schur decompositions of a tensor. Further, Stanimirovic et al. stan () introduced some basic properties of the range and null space of tensors, and the adequate definition of the tensor rank (i.e., reshaping rank). In view of reshape rank, Behera et al. behera18 () discussed full rank decomposition of tensors via Einstein product. The vast work on the generalized inverse of tensors bm (); ji2017 (); jin2017 (); stan (); sun () and its applications to the solution of multilinear systems bral (); Wei18 (), motivate us to study the characterizations of the Drazin inverse, W-weighted Drazin inverse and iterative technique in the framework of tensors.

In this paper, we further study the Drazin inverse of tensors. This study can lead to the enhancement of the computation of the Drazin inverse of tensors along with solutions of multilinear structure in multidimensional systems. In this regard, we discuss different characterizations of the Drazin inverse and W-weighted Drazin inverse of tensors. In addition to these, some new methods for computing the Drazin inverse of a tensor is proposed. Since the reduction of spatial dimensions and the generalized inverse of tensors needs to solve tensor-based partial differential equations, here we concentrate on the tensor iterative method (higher order Gauss-Seidel) and its convergence analysis using the theory of Einstein product.

1.1 Organization of the paper

The rest of the paper is organized as follows. In Section 2, we discuss some notations and definitions, which are the necessary ingredient for proving the main results in Sections 3-5. Several characterizations of the Drazin inverse are discussed in Section 3. Besides these, we have computed the Drazin inverse of tensors with the help of other generalized inverses. The notion of W-weighted Drazin inverse and a few properties of this inverse are introduced in Section 4. Then taking advantage of the Drazin inverse of tensors, we discuss the solution of multilinear systems in Section 5. In addition to these, the convergence analysis of the iterative technique is also investigated within the framework of the Einstein product. In Section 6, we conclude this paper with some remarks.

2 Preliminaries

2.1 Some notations and definitions

For convenience, we first briefly explain some of the terminologies which will be used here onwards. Let be the set of order and dimension tensors over the real (complex) field , where are positive integers. An order tensor is denoted as . Note that throughout the paper, tensors are represented in calligraphic letters like , and the notation represents the scalars. We use some additional notations to simplify our representation,

 j(N)={j1,j2,⋯,jN | 1≤jk≤Jk,k=1,2,⋯,N},  and   J(N)=J1×J2×⋯×JN.

Further, we denote . Let the tensor . In connection with these notations, the tensor and , is denoted by . The Einstein product ein () of tensors and is defined by the operation via

 (A∗NB)i(N),j(N)=∑k1...kNai1...iNk1...kNbk1...kNj1...jN=∑k(N)ai(N),k(N)bk(N),j(N).

In particular, if , then and

 (A∗NB)i(N)=∑k(N)ai(N),k(N)bk(N).

The Einstein product is used in the study of the theory of relativity ein () and in the area of continuum mechanics (lai ()). Using such theory of Einstein product, the range space and null space of a tensor was introduced in Wei18 (); stan (), as follows.

 R(A)={A∗NX: X∈RJ(N)} and N(A)={X: A∗NX=O∈RI(M)},

where the tensor denotes the zero tensor, i.e., all the entries are zero. As a consequence of the definition of the range space and null space of a tensor, it is clear that is a subspace of and is a subspace of . The relation of range space and some properties of range and null spaces are discussed in Wei18 (); stan (). Here, we collect some known results which will be used in this paper.

Lemma 2.1 (Lemma 2.2, stan ()).

Let , Then if and only if there exists such that

Adopting the definition of range space and null space, Ji and Wei Wei18 () discussed the index of a tensor, as follows.

Definition 2.2.

Wei18 () Let and k be the smallest nonnegative integer such that, Then is called the index of and denoted by

We now move to the definition of the Drazin inverse of a tensor, which was studied in Theorem 3.3 Wei18 () in the context of range space of a tensor, as follows.

Definition 2.3.

Let be a tensor with . The tensor satisfying the following three tensor equations:

 (1) Ak+1∗NX=Ak, (2) X∗NA∗NX=X, (3) A∗NX=X∗NA,

is called the Drazin inverse of , and is denoted by . Specifically, when k = 1, is called the group inverse of and denoted by .

At the same time, the authors of Wei18 () discussed the existence of the Drazin inverse of a tensor. In view of this, we have the following result for the uniqueness of the Drazin inverse of a tensor.

Theorem 2.4.

Let be a tensor with , then the Drazin inverse is unique.

Proof.

Suppose and are two Drazin inverses of . Now

 X=X∗NA∗NX=X∗NA∗NX∗NA∗NX=X∗NX∗NA∗NA∗NX=X2∗NA2∗NX. (2.1)

Repeating times the Eq. (2.1), we obtain

 X = Xk+1∗NAk+1∗NX=Xk+1∗NAk=Xk+1∗NAk∗NY∗NA = Xk+1∗NAk∗NY∗NA∗NY∗NA=Xk+1∗NAK+1∗NY2∗NA = Ak+1∗NXk+1∗NY2∗NA=A∗NX∗NY∗NA∗NY = A2∗NX∗NY2=A2∗NX∗NY∗NY.

Now substituting as and repeating it times, one can obtain . ∎

In connection with range space and null space, Ji and Wei Wei18 () discussed the characterization of the Drazin inverse of tensors, as follows.

Theorem 2.5 (Theorem 3.4, Wei18 ()).

Let and ind Then for the following holds

1. and

2. , , and .

Theorem 2.6 (Theorem 3.2, Wei18 ()).

Let . If ind, then and for any positive integer .

We next present the definition of the diagonal tensor which was introduced earlier in bral (); sun ().

Definition 2.7 (Diagonal tensor, Definition 3.12, bral ()).

A tensor is called a diagonal tensor if all its entries are zero except for

The definition of an upper off-diagonal tensor and lower off-diagonal tensor are defined under the influence of Definition 3.12, sun (), as follows.

Definition 2.8 (Upper off-diagonal tensor).

A tensor is called an upper off-diagonal tensor if all entries below the main diagonal are zero, i.e., = 0 for , where

Definition 2.9 (Lower off-diagonal tensor).

A tensor is called a lower off-diagonal tensor if all entries above the main diagonal are zero, i.e., = 0 for , where

Using the notation of diagonal tensor we define diagonal dominant tensor, as follow.

Definition 2.10 (Diagonally dominant tensor).

A tensor is called diagonally dominant if

 |ai(N),i(N)|≥∑j(N)≠i(N)j(N)|ai(N),j(N)|. (2.2)

We recall the definition of an eigenvalue of a tensor as below.

Definition 2.11 (Eigenvalue of a tensor, Definition 2.3, liang2019 ()).

Let A complex number is called an eigenvalue of if there exist some nonzero tensor such that

The nonzero tensor is called eigen vector of and we define the spectral radius of be the largest absolute value of the eigenvalues of . As a consequence of the definition of eigenvalue, the following lemma easily holds.

Lemma 2.12.

Let If is an eigenvalue of then for , is an eigenvalue of

Let . Then we have In view of this fact, we next present the definition of the convergence of a tensor.

Definition 2.13 (Convergent tensor).

A tensor is called convergent tensor if as .

We now introduce the definition of convergence of a power series of tensor, which is a generalization of the power series in matrices makar ().

Definition 2.14 (Tensor series convergent).

Let The series is convergent if is convergent for every and .

2.2 Reshape rank and decomposition of a tensor

The reshape operation systematically rearranges the entries of an arbitrary order-tensor into a matrix stan (). This operation is denoted by rsh, and implemented by means of the standard Matlab function reshape.

Definition 2.15 (Definition 3.1, stan ()).

The 1-1 and onto reshape map, rsh, is defined as with

 rsh(A)=A=reshape(A,M,N), (2.3)

where the matrix and . Further, the inverse reshaping is the mapping defined as with

 rsh−1(A)=A=reshape(A,I1,⋯,IM,J1,⋯,JN), (2.4)

where the matrix and the tensor .

Further, Lemma 3.2 in stan () defined the rank of a tensor , denoted by , as

 rshrank(A)=rank(rsh(A)). (2.5)

Adopting the reshaping operation, Behera et al. behera18 () defined the Moore-Penrose inverse of an arbitrary order tensor. Whereas, the authors of sun () was introduced this Moore-Penrose inverse for even-order tensors, which is recalled next.

Definition 2.16 (Definition 2.2, sun ()).

Let . The tensor satisfying the following four tensor equations:

 A∗NX∗NA=A;  X∗NA∗NX=X;  (A∗NX)∗=A∗NX;  (X∗NA)∗=X∗NA

is called the Moore-Penrose inverse of , and is denoted by . In particular, if the tensor satisfies only first equation, then is called -inverse of and denoted by

On the other hand, using reshape rank of a tensor , Behera et al. behera18 () discussed full rank decomposition of a tensor, as stated below.

Theorem 2.17.

[Theorem- 2.22 behera18 ()] Let . Then there exist a left invertible tensor and a right invertible tensor such that

 A=F∗RG, (2.6)

where .

In connection with the Moore-Penrose inverse of tensors, the singular value decomposition (SVD) discussed in Lemma 3.1 sun () for a complex tensor. However, the authors of bral () proved the same result for a real tensor.

Lemma 2.18.

(Lemma 3.1, sun ()) A tensor can be decomposed as

 A=U∗NB∗NV∗,

where and are unitary tensors, and is a tensor such that , if , where

3 Results on Drazin inverse of tensors

The Drazin inverse of tensors plays nearly the same role as the standard inverse of an invertible tensor. General properties of the Drazin inverse of tensors via Einstein product can be found in Wei18 (). In this section, we further embellish on this theory by producing a few more characterizations of this inverse. We divided this section into two parts. In the first part, we obtain several identities involving the Drazin inverse of tensors. The second part contains the computation of the Drazin inverse of tensors.

3.1 Some identities

It is worth mentioning that Ji and Wei Wei18 () studied the Drazin inverse of tensors, which motivates us to investigate further on the theory of the Drazin inverse of tensors. We find some interesting identities. Some of these are used in the next sections. The very first result of this section will numerously use in other consequential identities.

Lemma 3.1.

Let be a tensor with . Then .

Proof.

Using the method as in the proof of the above Lemma 3.1, one can show the next theorem.

Theorem 3.2.

Let be a tensor with . Then the following holds

1. If , then

2. If and , then

Recall that a tensor is called nilpotent if where is the zero tensor. It is trivial that, the nilpotent tensors are always singular. The next result presents the existence of the Drazin inverse of nilpotent tensors.

Corollary 3.3.

Let be a nilpotent tensor with index Then .

The power of the Drazin inverse and the Drazin inverse of power tensors can be switched without changing the result. Which is discussed in the following theorem.

Theorem 3.4.

Let and ind Then for the following holds

Proof.

Let the tensor . It is enough to show is the Drazin inverse of Now

further,

and

 (3.3)

From (3.1), (3.1) and (3.3), we conclude is the Drazin inverse of Hence
Let . By Definition 2.3, we have

further,

and

Thus .
Consider . Now applying the Definition of the Drazin inverse, we obtain

Further, we have . Hence is the Drazin inverse of . ∎

Corollary 3.5.

Let be a tensor with . Then the following holds

1. for

2. .

Remark 3.6.

Like the well-known property of the Moore Penrose inverse, i.e., (see Proposition 3.3, sun ()), it is worth pointing out that the Drazin inverse is not following the property, i.e., , as shown below with an example.

Example 3.7.

Consider a tensor with entries

 aij11=(011111), aij12=(001011), aij13=(000001), aij21=(011011), aij22=(001001), aij23=(000000).

Then one can easily verify , however .

At this point one may be interested to know, when does the above equality property (Remark 3.6) hold The answer to this query is discussed in the next theorem.

Theorem 3.8.

Let . Then if and only if ind

Proof.

Let be of index . Now . Since is of index so and . Conversely, let . By definition of the Drazin inverse, . This implies . So by Lemma 2.1, It is obvious that Thus . Hence is of index . ∎

Thus, the special case of the Drazin inverse (index = 1) gives necessary and sufficient conditions for the equality.

Remark 3.9.

The Theorem 3.4 (a) is not true if we use two different tensor and , i.e., , where and .

Example 3.10.

Consider the tensor defined in Example (3.7) and a tensor with entries

 bij11=(000000), bij12=(100100), bij13=(110110), bij21=(100000), bij22=(110100), bij23=(111110).

Then and with entries

 xij11=(000000), xij12=(120−1−10), xij13=(0020−1−1), xij21=(0−10200), xij22=(0−1−1020), xij23=(00−1001).

Hence .

The above remark (3.9) can be stated as reverse order law for the Drazin inverse of tensors, which is a fundamental in the theory of generalized inverses of tensors. Recently, there has been increasing interest in studying reverse order law of tensors based on different generalized inverses behera18 (); Mispa18 (); panigrahy (). In this regard, we discuss one sufficient condition of the reverse order law for the Drazin inverse of tensors, as follows.

Theorem 3.11.

Suppose are each of index . If , then the followings are true

1. and

2. .

Since