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The dual index and dual core generalized inverse

  • Hongxing Wang and Ju Gao
Published/Copyright: July 6, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we introduce the dual index and dual core generalized inverse (DCGI). By applying rank equation, generalized inverse, and matrix decomposition, we give several characterizations of the dual index when it is equal to 1. We realize that if DCGI exists, then it is unique. We derive a compact formula for DCGI and a series of equivalent characterizations of the existence of the inverse. It is worth noting that the dual index of A ^ is equal to 1 if and only if its DCGI exists. When the dual index of A ^ is equal to 1, we study dual Moore-Penrose generalized inverse (DMPGI) and dual group generalized inverse (DGGI) and consider the relationships among DCGI, DMPGI, DGGI, Moore-Penrose dual generalized inverse, and other dual generalized inverses. In addition, we consider symmetric dual matrix and its dual generalized inverses. Finally, two examples are given to illustrate the application of DCGI in linear dual equations.

Keywords: dual core generalized inverse; dual index; dual Moore-Penrose generalized inverse; dual group generalized inverse; Moore-Penrose dual generalized inverse; dual analog of least-squares solutions
MSC 2010: 15A09; 15A66; 65F05

1 Introduction

The concept of dual number was introduced by Clifford in 1873 [1], and the name of the number is given by Study in 1903 [2]. The dual number consists of a real unit 1 and a Clifford operator ε . The dual number contains two real elements, i.e., a ^ = a + ε a ′ , where the real elements a and a ′ are called the real part and dual part of a ^ , respectively. The rule is ε ≠ 0 , 0 ε = ε 0 = 0 , 1 ε = ε 1 = ε , and ε 2 = 0 . When we discuss the geometry of directed lines in space, we can take the angle “ θ ” as the real part and the vertical distance “ s ” as the dual part to form the dual angle, θ ^ = θ + ε s . As an extension of the concept of dual number, dual vector is to replace the real elements of a real vector with dual numbers and is often used as a mathematical expression for helices. A matrix with dual numbers as elements is called a dual matrix. Denote an m × n dual matrix as A ^ , which is represented as follows:

(1.1) A ^ = A + ε B ,

in which A ∈ R m , n and B ∈ R m , n . The matrix A ( B ) is called the real(dual) part of the dual matrix A ^ . The symbol D m , n denotes the set of all m × n dual matrices; I m is an m -order identity matrix; ℛ ( A ^ ) represents the range of A ^ . Furthermore, denote A ^ T = A T + ε B T . When A ^ T = A ^ , we say that A ^ is symmetric.

Dual matrices are used in many fields today. In Kinematics, e.g., with the aid of the principle of transference [3], many problems can be initially stated under the condition of spherical motion and then extended to spiral motion after the dualization of the equation, which makes the dual matrix widely used in space agency kinematics analysis and synthesis [4–8] and robotics [9–13]. Their presence is also felt in other areas of science and engineering, which has raised interest in various aspects of linear algebra and computational methods associated with their use [14–18]. Keler [19], Beyer [20], and others have done pioneering work in the engineering applications of dual algebra.

Pennestrì and Valentini [13] introduced the Moore-Penrose dual generalized inverse (MPDGI): let A ^ = A + ε B , then the MPDGI of A ^ is denoted by A ^ P and is displayed in the form A ^ P = A + − ε A + B A + . For a given dual matrix A ^ , if there exists a dual matrix X ^ satisfying

( 1 ) A ^ X ^ A ^ = A ^ , ( 2 ) X ^ A ^ X ^ = X ^ , ( 3 ) ( A ^ X ^ ) T = A ^ X ^ , ( 4 ) ( X ^ A ^ ) T = X ^ A ^ ,

then we call X ^ the dual Moore-Penrose generalized inverse (DMPGI) of A ^ , and denote it as A ^ + [17]. It is worth noting that for any dual matrix, its MPDGI always exists, while its DMPGI may not exist. Furthermore, if X ^ satisfies A ^ X ^ A ^ = A ^ , we call X ^ a { 1 } -dual generalized inverse of A ^ and denote it as A ^ ( 1 ) ; if X ^ satisfies A ^ X ^ A ^ = A ^ and ( A ^ X ^ ) T = A ^ X ^ , we call X ^ a { 1 , 3 } -dual generalized inverse of A ^ and denote it as A ^ ( 1 , 3 ) . Recently, Wang has given some necessary and sufficient conditions for a dual matrix to have the DMPGI, and some equivalent relations between the DMPGI and the MPDGI in [21].

Lemma 1.1

[21] Let A ^ = A + ε B ∈ D m , n , then the following conditions are equivalent:

  1. The DMPGI A ^ + of A ^ exists;

  2. ( I m − A A + ) B ( I n − A + A ) = 0 ;

  3. rank B A A 0 = 2 rank ( A ) .

Furthermore, when the DMPGI A ^ + of A ^ exists,

(1.2) A ^ + = A + − ε ( A + B A + − ( A T A ) + B T ( I m − A A + ) − ( I n − A + A ) B T ( A A T ) + ) .

Lemma 1.2

[21] Let A ^ = A + ε B ∈ D m , n , then the DMPGI A ^ + of A ^ exists, and A ^ + = A ^ P if and only if ( I m − A A + ) B = 0 and B ( I n − A + A ) = 0 .

MPDGI and DMPGI are used in many aspects. For example, Pennestrì and Valentini, in their study [13], applied MPDGI to various motions such as rigid body translation and dual angular velocity acquisition. In a study by Pennestrì et al. [17], DMPGI is applied to kinematic synthesis of spatial mechanisms, and a series of numerical examples about calculation and application of DMPGI to kinematic synthesis of linkage mechanisms is given. In addition, MPDGI are also used in many inverse problems of kinematics and analysis of machines and mechanisms in a study by de Falco et al. [4].

Next, Zhong and Zhang [22] introduced the dual group generalized inverse (DGGI): let A ^ be an n -square dual matrix. If there exists an n -square dual matrix G ^ satisfying

( 1 ) A ^ G ^ A ^ = A ^ , ( 2 ) G ^ A ^ G ^ = G ^ , ( 5 ) A ^ G ^ = G ^ A ^ ,

then A ^ is called a dual group generalized invertible matrix, and G ^ is the DGGI of A ^ , which is recorded as A ^ # . Zhong and Zhang [22] gave some necessary and sufficient conditions for a dual matrix to have DGGI and apply DGGI to study linear dual equations.

Lemma 1.3

[22] Let A ^ = A + ε B be a dual matrix with A , B ∈ R n , n and Ind ( A ) = 1 , then the DGGI of A ^ exists if and only if ( I n − A A # ) B ( I n − A A # ) = 0 .

Furthermore, if the dual group inverse of A ^ exists, then

(1.3) A ^ # = A # + ε R ,

where

(1.4) R = − A # B A # + ( A # ) 2 B ( I n − A A # ) + ( I n − A A # ) B ( A # ) 2 .

Most applications of dual algebra in Kinematics require numerical solutions to linear dual equations. Udwadia [23] introduced the norm of dual vector and used some properties of dual generalized inverses in solving linear dual equations. Various dual generalized inverses are useful for solving consistent linear dual equations or inconsistent linear dual equations. In particular, Qi et al., in their study [24], proposed both a total order and an absolute value function for dual numbers. Then, they gave the definition of the magnitude of a dual quaternion as a dual number. Furthermore, 1-norm, ∞ -norm, and 2-norm are extended to dual quaternion vectors in their article. Furthermore, Qi et al. began to study a series of basic problems in [24–28], e.g., singular value decomposition of dual complex matrices, low rank approximation of dual complex matrices, dual quaternion vectors, generalized inverses of dual quaternion matrices, and others.

It is known that Moore-Penrose inverse and group inverse belong to generalized inverses in complex fields. Other well-known generalized inverses are Drazin inverse, core inverse, and so on. The core inverse means that when the index of A ∈ R n , n is 1, there is a unique matrix X ∈ R n , n , which satisfies A X A = A , A X 2 = X and ( A X ) T = A X . We call it the core inverse of matrix A , which is expressed as A #⃝ . Baksalary and Trenkler, in their study [29], obtained A #⃝ = A # A A + . The core inverse has good properties. It can be used to solve many problems, especially in constrained least squares problem.

Although DMPGI, MPDGI, and DGGI are discussed in [21,22], the dual core generalized inverse (DCGI) and the dual index have not been studied yet. On the basis of the above studies, the concepts of the dual index and DCGI are introduced in this article. Furthermore, it is proved that when the dual index of dual matrix is 1, there must be the DCGI of the dual matrix. The sufficient and necessary condition, namely the index is 1, is used to identify the existence of DCGI, which makes the problem more concise and clear. At the same time, we also give other equivalent conditions for the existence of DCGI and the compact formula for DCGI, as well as the relations among DCGI, DGGI, DMPGI, and MPDGI of the dual matrix, and discuss some special dual matrices. Finally, we solve two linear dual equations by applying DCGI.

2 Dual index 1

In complex (real) field, the index is necessary for studying generalized inverse and its related problems. For example, it is known that the group (core) inverse of a matrix exists if and only if the index of the matrix is equal to 1. Moreover, Wei et al. [30–32] considered singular linear structured system with index 1. In this section, we introduce the dual index of a dual matrix. We provide some necessary and sufficient conditions for a dual matrix with the dual index of 1. Furthermore, by using the dual index, we study the dual group generalized invertible matrix.

Definition 2.1

Let A ^ be an n -square dual matrix. If ℛ ( A ^ 2 ) = ℛ ( A ^ ) , then the dual index of A ^ is equal to 1, and it is recorded as Ind ( A ^ ) = 1 .

Next, we discuss the equivalent characterization of ℛ ( A ^ 2 ) = ℛ ( A ^ ) , where A ^ = A + ε B . ℛ ( A ^ 2 ) ⊆ ℛ ( A ^ ) is constant, which means that ℛ ( A ^ 2 ) = ℛ ( A ^ ) if and only if ℛ ( A ^ ) ⊆ ℛ ( A ^ 2 ) , i.e., there exists X ^ = X 1 + ε X 2 , which makes

A ^ 2 X ^ = A ^ .

Put A ^ = A + ε B and X ^ = X 1 + ε X 2 into the above equation to obtain

A + ε B = ( A + ε B ) 2 ( X 1 + ε X 2 ) = A 2 X 1 + ε ( A 2 X 2 + ( A B + B A ) X 1 ) ,

i.e.,

A 2 X 1 = A , (2.1a) A 2 X 2 + ( A B + B A ) X 1 = B . (2.1b)

From equation (2.1a), we see that A 2 X 1 = A is consistent if and only if

(2.2) rank ( A ) = rank ( A 2 ) ,

i.e., the index of real matrix A is equal to 1. It is obvious that ( A 2 ) # = ( A # ) 2 and A 2 ( A # ) 2 = A A # . Therefore, we can obtain the general solution to equation (2.1a) as follows:

(2.3) X 1 = ( A 2 ) # A + ( I n − A 2 ( A # ) 2 ) Y = A # + ( I n − A # A ) Y ,

where Y is arbitrary.

By substituting equation (2.3) into equation (2.1b), we obtain B = A 2 X 2 + ( A B + B A ) ( A # + ( I n − A # A ) Y ) . From A ( A # + ( I n − A # A ) Y ) = A A # , it follows that B = A 2 X 2 + A B ( A # + ( I n − A # A ) Y ) + B A A # = A 2 X 2 + A B A # + A B ( I n − A # A ) Y + B A A # . Therefore,

(2.4) A 2 A B ( I n − A # A ) X 2 Y = B − A B A # − B A A # .

By applying, (2.2), we obtain (2.4) if and only if

(2.5) rank A 2 A B ( I n − A # A ) B − A B A # − B A A # = rank A 2 A B ( I n − A # A ) .

Since

A 2 A B ( I n − A # A ) I n − A # B ( I n − A # A ) 0 I n = A 2 0 ,

by applying equation (2.2), we obtain

rank ( A 2 ) = rank ( A ) = rank A 2 A B ( I n − A # A ) B − ( A B + B A ) A # = rank A B − ( A B + B A ) A # = rank A B − A B A # − B A A # = rank A B − B A A # = rank A B ( I n − A A # ) .

Then, the consistency of equation (2.4) is equivalent to

(2.6) rank ( A ) = rank A B ( I n − A A # ) .

According to equations (2.2) and (2.6), the dual index of A ^ is equal to 1, which is equivalent to

(2.7) rank ( A 2 ) = rank ( A ) = rank A B ( I n − A A # ) .

Therefore, we have the following theorem.

Theorem 2.1

Let A ^ = A + ε B ∈ D n , n and rank ( A ) = r , then the dual index of A ^ is equal to 1, which is equivalent to Ind ( A ) = 1 , and

(2.8) rank ( A ) = rank A B ( I n − A A # ) .

Next, we present a well-known matrix decomposition [33, Corollary 6] and several corresponding decompositions of generalized inverses, which will be used in the following part of this article. Let A ∈ R n , n with rank ( A ) = r , then

(2.9) A = U Σ K Σ L 0 0 U T ,

where U ∈ R n , n is unitary, nonsingular Σ = diag ( σ 1 , … , σ r ) is the diagonal matrix of singular values of A , σ 1 ≥ ⋯ ≥ σ r > 0 , and K ∈ R r , r and L ∈ R r , n − r satisfy

(2.10) K K T + L L T = I r .

By applying the decomposition, Baksalary and Trenkler [29] obtain

(2.11) A + = U K T Σ − 1 0 L T Σ − 1 0 U T .

Especially, when the index of A is 1, the necessary and sufficient condition for the existence of A #⃝ is that K is nonsingular. In [29], by applying equation (2.9) Baksalary and Trenkler also give characterizations of core inverse and group inverse:

(2.12) A #⃝ = A # A A +

(2.13) = U ( Σ K ) − 1 0 0 0 U T ,

(2.14) A # = U K − 1 Σ − 1 K − 1 Σ − 1 K − 1 L 0 0 U T .

Based on the premise that the index of A ∈ R n , n is 1, we analyze equation (2.8). Let the decomposition of A be of the form in equation (2.9). Then, we write

(2.15) B = U B 1 B 2 B 3 B 4 U T ,

where B 1 is an r -square matrix and r = rank ( A ) . By substituting equations (2.9), (2.14), and (2.15) into B ( I n − A A # ) , we obtain

B ( I n − A A # ) = U 0 − B 1 K − 1 L + B 2 0 − B 3 K − 1 L + B 4 U T .

It follows from equations (2.8) and (2.9) that

rank Σ K Σ L 0 0 = rank Σ K Σ L 0 − B 1 K − 1 L + B 2 0 0 0 − B 3 K − 1 L + B 4 ,

which implies that B 4 = B 3 K − 1 L .

In summary, we obtain that rank ( A ) = rank A B ( I n − A A # ) if and only if B 4 = B 3 K − 1 L . Therefore, we have the following Theorem 2.2.

Theorem 2.2

Let A ^ = A + ε B ∈ D n , n , rank ( A ) = r , and A and B have the forms as in equations (2.9) and (2.15), respectively. Then, the dual index of A ^ is equal to 1, which is equivalent to Ind ( A ) = 1 and B 4 = B 3 K − 1 L .

Furthermore, since Ind ( A ) = 1 , applying equations (2.9) and (2.15), it is easy to check that

rank B A A 0 = rank B 1 B 2 Σ K Σ L B 3 B 4 0 0 Σ K Σ L 0 0 0 0 0 0 = 2 rank ( A ) + rank ( B 4 − B 3 K − 1 L ) .

By applying Theorem 2.2, we have the following Theorem 2.3.

Theorem 2.3

Let A ^ = A + ε B ∈ D n , n and rank ( A ) = r , then the dual index of A ^ is equal to 1, which is equivalent to Ind ( A ) = 1 , and

rank B A A 0 = 2 rank ( A ) .

In the following theorems, we give some equivalent characterizations with dual index 1.

Theorem 2.4

Let A ^ = A + ε B ∈ D n , n , where rank ( A ) = r , then the dual index of A ^ is 1 if and only if Ind ( A ) = 1 and ( I n − A A + ) B ( I n − A + A ) = 0 .

Proof

By applying equations (2.9) and (2.11), we obtain

(2.16) I n − A A + = U 0 0 0 I n − r U T ,

(2.17) I n − A + A = U I r − K T K − K T L − L T K I n − r − L T L U T .

Then, by equation (2.15)

(2.18) ( I n − A A + ) B ( I n − A + A ) = U 0 0 B 3 − B 3 K T K − B 4 L T K − B 3 K T L + B 4 − B 4 L T L U T .

Let the dual index of A ^ be equal to 1. According to Theorem 2.2, the index of A is 1 and B 4 = B 3 K − 1 L . Substituting B 4 = B 3 K − 1 L into B 3 − B 3 K T K − B 4 L T K and − B 3 K T L + B 4 − B 4 L T L and applying equation (2.10), we obtain

B 3 − B 3 K T K − B 4 L T K = B 3 − B 3 K T K − B 3 K − 1 L L T K = B 3 − B 3 K T K − B 3 K − 1 K + B 3 K − 1 K K T K = B 3 − B 3 K T K − B 3 + B 3 K T K = 0

and

− B 3 K T L + B 4 − B 4 L T L = − B 3 K T L + B 3 K − 1 L − B 3 K − 1 L L T L = − B 3 K T L + B 3 K − 1 L − B 3 K − 1 L + B 3 K − 1 K K T L = − B 3 K T L + B 3 K − 1 L − B 3 K − 1 L + B 3 K T L = 0 .

Therefore, from equation (2.18), it follows that ( I n − A A + ) B ( I n − A + A ) = 0 .

Conversely, let the index of A is 1 and ( I n − A A + ) B ( I n − A + A ) = 0 . Applying equation (2.18) gives

B 3 − B 3 K T K − B 4 L T K = 0 , − B 3 K T L + B 4 − B 4 L T L = 0 ,

i.e.,

B 4 L T K = B 3 − B 3 K T K , (2.20a) B 3 K T L = B 4 − B 4 L T L . (2.20b)

Since the index of A is 1, it is known that K is a nonsingular matrix. Post-multiplying both sides of equation (2.20a) by K − 1 L , we obtain

(2.21) B 3 K − 1 L − B 3 K T L = B 4 L T L .

By substituting equation (2.20b) into (2.21), we obtain B 3 K − 1 L − B 4 + B 4 L T L = B 4 L T L , i.e., B 4 = B 3 K − 1 L . In summary, the index of A is 1 and B 4 = B 3 K − 1 L . It follows from Theorem 2.2 that the dual index of A ^ is 1.□

Theorem 2.5

Let A ^ = A + ε B ∈ D n , n , where rank ( A ) = r , then the dual index of A ^ is 1 if and only if Ind ( A ) = 1 and ( I n − A A # ) B ( I n − A A # ) = 0 .

Proof

Let Ind ( A ) = 1 . By applying equations (2.9), (2.14), and (2.15), we can obtain

(2.22) I n − A A # = U 0 − K − 1 L 0 I n − r U T , ( I n − A A # ) B ( I n − A A # ) = U 0 K − 1 L B 3 K − 1 L − K − 1 L B 4 0 − B 3 K − 1 L + B 4 U T .

If the dual index of A ^ is 1, from Theorem 2.2, we can obtain the index of A is 1 and B 4 = B 3 K − 1 L . Therefore, K − 1 L B 3 K − 1 L − K − 1 L B 4 = 0 and − B 3 K − 1 L + B 4 = 0 . It follows from equation (2.22) that ( I n − A A # ) B ( I n − A A # ) = 0 .

Conversely, let the index of A is 1 and ( I n − A A # ) B ( I n − A A # ) = 0 . Applying equation (2.22) gives B 4 = B 3 K − 1 L . To sum up, the index of A is 1 and B 4 = B 3 K − 1 L . Furthermore, according to Theorem 2.2, we obtain that the dual index of A ^ is 1.□

By applying Lemma 1.3 and Theorem 2.5, we obtain the following Theorem 2.6 that discusses the relationship between DGGI and dual index 1.

Theorem 2.6

Let A ^ = A + ε B ∈ D n , n , then the dual index of A ^ is 1 if and only if A ^ # exists.

Proof

From Lemma 1.3, we see that the DGGI of A ^ exists if and only if Ind ( A ) = 1 and ( I n − A A # ) B ( I n − A A # ) = 0 . From Theorem 2.5, we see that the dual index of A ^ is 1 if and only if Ind ( A ) = 1 and ( I n − A A # ) B ( I n − A A # ) = 0 . Therefore, we obtain that the dual index of A ^ is 1 if and only if A ^ # exists.□

By applying Theorem 2.4, we see that the dual index of A ^ is 1 if and only if Ind ( A ) = 1 and ( I n − A A + ) B ( I n − A + A ) = 0 . By applying Lemma 1.1, we see that the DMPGI A ^ + of A ^ exists if and only if ( I m − A A + ) B ( I n − A + A ) = 0 . Therefore, we obtain the relationship between dual index 1 and DMPGI in the following Theorem 2.7.

Theorem 2.7

Let A ^ = A + ε B ∈ D n , n , then the dual index of A ^ is 1 if and only if Ind ( A ) = 1 and A ^ + exists.

Theorem 2.8

Let A ^ = A + ε B ∈ D n , n and Ind ( A ) = 1 , then A ^ + exists if and only if rank ( A ) = rank A B ( I n − A A # ) .

Proof

“ ⇒ ” If A ^ + exists, then rank B A A 0 = 2 rank ( A ) is known by the Lemma 1.1. Therefore, when the index of A is 1, the dual index of A ^ is equal to 1 from Theorem 2.3. It is also known from Theorem 2.1 that rank ( A ) = rank A B ( I n − A A # ) .

“ ⇐ ” Let rank ( A ) = rank A B ( I n − A A # ) . When the index of A is 1, the dual index of A ^ is equal to 1 from Theorem 2.1. So rank B A A 0 = 2 rank ( A ) from Theorem 2.3. By conditions (i) and (iii) of Lemma 1.1, we know that A ^ + exists.□

3 DCGI

It is well known that a matrix is group invertible if and only if its index is 1 and is core invertible in R n , n . In Section 2, we obtain that the dual index of A ^ = A + ε B is 1 if and only if A ^ # exists. In this section, we introduce DCGI, give some properties and characterizations of the inverse, and consider relationships among DCGI, DGGI, and dual index 1. Meanwhile, we also give characterizations of some other interesting dual generalized inverses.

3.1 Definition and uniqueness of DCGI

Definition 3.1

Let A ^ be an n -square dual matrix. If there exists an n -square dual matrix G ^ satisfying

(3.1) ( 1 ) A ^ G ^ A ^ = A ^ , ( 2 ′ ) A ^ G ^ 2 = G ^ , ( 3 ) ( A ^ G ^ ) T = A ^ G ^ ,

then A ^ is called a dual core generalized invertible matrix, and G ^ is the DCGI of A ^ , which is recorded as A ^ #⃝ .

Theorem 3.1

Let A ^ = A + ε B ∈ D n , n , then the existence of the DCGI of A ^ is equivalent to the existence of G and R , which meet the following requirements: G = A #⃝ and

B G A + A R A + A G B = B , (3.2a) A G R + A R G + B G 2 = R , (3.2b) ( A R + B G ) T = A R + B G . (3.2c)

Furthermore, G ^ = G + ε R is the DCGI of A ^ .

Proof

From A ^ = A + ε B , G ^ = G + ε R , and

A ^ G ^ A ^ = ( A + ε B ) ( G + ε R ) ( A + ε B ) = A G A + ε ( B G A + A R A + A G B ) , A ^ G ^ 2 = ( A + ε B ) ( G + ε R ) 2 = A G 2 + ε ( A G R + A R G + B G 2 ) , ( A ^ G ^ ) T = ( ( A + ε B ) ( G + ε R ) ) T = ( A G ) T + ε ( A R + B G ) T ,

we obtain that A ^ G ^ A ^ = A ^ , A ^ G ^ 2 = G ^ , and ( A ^ G ^ ) T = A ^ G ^ are, respectively, equivalent to

A G A = A , B G A + A R A + A G B = B , A G 2 = G , A G R + A R G + B G 2 = R , ( A G ) T = A G , ( A R + B G ) T = A R + B G .

Since A G A = A , A G 2 = G , and ( A G ) T = A G , we have G = A #⃝ . Therefore, if the DCGI of A ^ exists and G ^ = G + ε R is the DCGI of A ^ , then G = A #⃝ and equations (3.2a), (3.2b) and (3.2c) are established.

Conversely, let G ^ = G + ε R satisfy equations (3.2a), (3.2b), (3.2c) and G = A #⃝ . By applying Definition 3.1, it is easy to check that G ^ is the DCGI of A ^ . So, the DCGI of A ^ exists.□

According to Theorem 3.1, we can see that the existence of the core inverse of A is only a necessary condition for the dual core generalized invertibility of A ^ , that is to say, even though the real part of a dual matrix is core invertible, it may be also a dual matrix without DCGI.

Example 3.1

Let

(3.3) A ^ = A + ε B = a 0 0 0 b 0 0 0 0 + ε b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 ,

where a , b , and b 33 are not 0 and b i j ( i = 1 , 2 , 3 , j = 1 , 2 ) , b 13 , and b 23 are arbitrary real numbers. It is obvious that the real part A is core invertible and

(3.4) A #⃝ = 1 a 0 0 0 1 b 0 0 0 0 .

Let G ^ = A #⃝ + ε R , where

(3.5) R = r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 .

According to Theorem 3.1, if G ^ is the DCGI of A ^ , then R is a suitable matrix of order n and satisfies equations (3.2a), (3.2b), and (3.2c). Equation (3.2a) requires

(3.6) Δ = B A #⃝ A ︸ S 1 + A R A ︸ S 2 + A A #⃝ B ︸ S 3 ︸ S 123 − B = 0 .

Now we will prove that for any A ^ constructed by A and B in equation (3.3), equation (3.6) does not satisfy any three-order matrix R . Therefore, we need to prove S 123 − B ≠ 0 . As shown below, from equations (3.3)–(3.5), we have

S 1 = B A #⃝ A = b 11 b 12 0 b 21 b 22 0 b 31 b 32 0 , S 2 = A A #⃝ B = b 11 b 12 b 13 b 21 b 22 b 23 0 0 0 , S 3 = A R A = a 2 r 1 a b r 2 0 a b r 4 b 2 r 5 0 0 0 0 , S 123 = S 1 + S 2 + S 3 = a 2 r 1 + 2 b 11 2 b 12 + a b r 2 b 13 2 b 21 + a b r 4 2 b 22 + b 2 r 5 b 23 b 31 b 32 0 , Δ = S 123 − B = a 2 r 1 + b 11 b 12 + a b r 2 0 b 21 + a b r 4 b 22 + b 2 r 5 0 0 0 − b 33 .

If b 33 is not 0, then Δ is not 0 no matter what the matrix R is. Therefore, the core inverse condition (3.2a) is not satisfied, and A ^ of the set has no DCGI.

In the following Theorem 3.2, we consider the uniqueness of DCGI.

Theorem 3.2

The DCGI of any dual matrix is unique if it exists.

Proof

Let A ^ = A + ε B ∈ D n , n , rank ( A ) = r , and A ^ #⃝ = A #⃝ + ε R . Suppose that T ^ is any DCGI of A ^ , from Theorem 3.1 and the uniqueness of the core inverse of a real matrix, we can denote T ^ as T ^ = A #⃝ + ε R ˜ . Furthermore, we write

X = R − R ˜ .

Next, we prove X = 0 .

From equation (3.2a), it can be seen that

(3.7) B = B A #⃝ A + A R A + A A #⃝ B , B = B A #⃝ A + A R ˜ A + A A #⃝ B .

Through the first equation minus the second equation in (3.7), we obtain

(3.8) 0 = A ( R − R ˜ ) A = A X A .

From equation (3.2b), we have

(3.9) R = A A #⃝ R + A R A #⃝ + B ( A #⃝ ) 2 , R ˜ = A A #⃝ R ˜ + A R ˜ A #⃝ + B ( A #⃝ ) 2 .

Through the first equation minus the second equation in (3.9), we have

(3.10) X = R − R ˜ = A A #⃝ X + A X A #⃝ .

Similarly, from equation (3.2c), it can be seen that R and R ˜ satisfy

(3.11) ( A R + B A #⃝ ) T = A R + B A #⃝ , ( A R ˜ + B A #⃝ ) T = A R ˜ + B A #⃝ .

Through the first equation minus the second equation in (3.11), we have ( A ( R − R ˜ ) ) T = A ( R − R ˜ ) , i.e.,

(3.12) ( A X ) T = A X .

By post-multiplying both sides of equation (3.8) by A #⃝ , and by applying equation (3.12), we obtain

0 = A X A = A X A A #⃝ = ( A X ) T ( A A #⃝ ) T = X T A T ( A #⃝ ) T A T = X T ( A A #⃝ A ) T = X T A T = ( A X ) T = A X ,

i.e., A X = 0 . Thus, the equation (3.10) is simplified as follows:

(3.13) X = R − R ˜ = A A #⃝ X .

Let the decomposition of A be as in equation (2.9). We write

(3.14) X = U X 1 X 2 X 3 X 4 U T ,

where X 1 ∈ R r , r . Substituting equations (2.9), (2.13), and (3.14) into equation (3.13), we obtain

A A #⃝ X = U Σ K Σ L 0 0 U T U ( Σ K ) − 1 0 0 0 U T U X 1 X 2 X 3 X 4 U T = U I r 0 0 0 U T U X 1 X 2 X 3 X 4 U T = U X 1 X 2 0 0 U T = U X 1 X 2 X 3 X 4 U T = X .

Therefore, X 3 = X 4 = 0 , i.e.,

(3.15) X = U X 1 X 2 0 0 U T .

To make X = 0 hold, we only need to prove X 1 = 0 and X 2 = 0 .

By substituting equations (2.9) and (3.15) into equation (3.8), we obtain

A X A = U Σ K Σ L 0 0 U T U X 1 X 2 0 0 U T U Σ K Σ L 0 0 U T = U Σ K X 1 Σ K X 2 0 0 U T U Σ K Σ L 0 0 U T = U Σ K X 1 Σ K Σ K X 1 Σ L 0 0 U T = 0 .

Therefore,

Σ K X 1 Σ K = 0 , (3.16a) Σ K X 1 Σ L = 0 . (3.16b)

Since A is core invertible, from Theorem 3.1, we see that K is nonsingular. Thus, from equations (3.15) and (3.16a), we obtain X 1 = 0 . So

(3.17) X = U 0 X 2 0 0 U T .

Similarly, by substituting equations (2.9) and (3.17) into equation (3.12), we obtain

U Σ K Σ L 0 0 0 X 2 0 0 U T T = U Σ K Σ L 0 0 0 X 2 0 0 U T ,

i.e.,

U 0 Σ K X 2 0 0 U T T = U 0 Σ K X 2 0 0 U T .

Continue to simplify the above equation and obtain

U 0 0 ( Σ K X 2 ) T 0 U T = U 0 Σ K X 2 0 0 U T .

Thus, Σ K X 2 = 0 . Considering that both Σ and K are nonsingular matrices, we obtain X 2 = 0 .

To sum up, we obtain X 1 = 0 , X 2 = 0 , X 3 = 0 , and X 4 = 0 . From equation (3.14), we obtain X = 0 , which can also be understood as

R = R ˜ .

Therefore, R satisfying equations (3.2a), (3.2b), and (3.2c) is unique, i.e., if the DCGI of A ^ exists, then the inverse is unique.□

3.2 Characterizations and properties of DCGI

Theorem 3.3

Let A ^ = A + ε B ∈ D n , n , then its DCGI exists if and only if its dual index is 1.

Proof

Suppose that A is core invertible. Let the decomposition of A be as in equation (2.9), and the form of B be as in equation (2.15). We write

(3.18) R = U R 1 R 2 R 3 R 4 U T ,

where R 1 is an r -square matrix and r = rank ( A ) .

“ ⇒ ” Assuming that the dual core inverse A ^ #⃝ = G + ε R of the dual matrix A ^ = A + ε B exists, it can be seen from Theorem 3.1 that the real part matrix A is core invertible and G = A #⃝ , so the index of A is 1.

Since the DCGI exists, then equation (3.2a) holds. Substituting equations (2.9), (2.13), (2.15), and (3.18) into B G A + A R A + A G B = B , we obtain

U B 1 B 2 B 3 B 4 U T = U B 1 + ( Σ K R 1 + Σ L R 3 ) Σ K + B 1 B 1 K − 1 L + ( Σ K R 1 + Σ L R 3 ) Σ L + B 2 B 3 B 3 K − 1 L U T .

So we have B 4 = B 3 K − 1 L .

To sum up, from Theorem 2.2, when the index of A is 1 and B 4 = B 3 K − 1 L , the dual index of A ^ is 1.

“ ⇐ ” Let the dual index of A ^ be 1. According to Theorem 2.2, the index of A is 1 and B 4 = B 3 K − 1 L . It follows from equation (2.15) that

(3.19) B = U B 1 B 2 B 3 B 3 K − 1 L U T .

From equations (3.19) and (2.9), we obtain

A ^ = A + ε B = U Σ K Σ L 0 0 U T + ε U B 1 B 2 B 3 B 3 K − 1 L U T .

Denote

(3.20) G ^ = G + ε R = U ( Σ K ) − 1 0 0 0 U T + ε U − K − 1 L B 3 ( Σ K ) − 2 − ( Σ K ) − 1 B 1 ( Σ K ) − 1 K − 1 Σ − 2 ( B 3 K − 1 ) T B 3 ( Σ K ) − 2 0 U T .

Then,

A ^ G ^ A ^ = A G A + ε ( B G A + A R A + A G B ) = U Σ K Σ L 0 0 U T + ε U B 1 B 1 ( Σ K ) − 1 Σ L B 3 B 3 ( Σ K ) − 1 Σ L U T + U − B 1 − B 1 ( Σ K ) − 1 Σ L 0 0 U T + U B 1 B 2 0 0 U T = U Σ K Σ L 0 0 U T + ε U B 1 B 2 B 3 B 3 K − 1 L U T = A + ε B = A ^ , A ^ G ^ 2 = A G 2 + ε ( A G R + A R G + B G 2 ) = U ( Σ K ) − 1 0 0 0 U T + ε U − K − 1 L B 3 ( Σ K ) − 2 − ( Σ K ) − 1 B 1 ( Σ K ) − 1 K − 1 Σ − 2 ( B 3 K − 1 ) T 0 0 U T + ε U − B 1 ( Σ K ) − 2 0 0 0 + B 1 ( Σ K ) − 2 0 B 3 ( Σ K ) − 2 0 U T = U ( Σ K ) − 1 0 0 0 U T + ε U − K − 1 L B 3 ( Σ K ) − 2 − ( Σ K ) − 1 B 1 ( Σ K ) − 1 K − 1 Σ − 2 ( B 3 K − 1 ) T B 3 ( Σ K ) − 2 0 U T = G + ε R = G ^

and

( A ^ G ^ ) T = ( A G + ε ( A R + B G ) ) T = U I r 0 0 0 U T T + ε U 0 Σ − 1 ( B 3 K − 1 ) T B 3 ( Σ K ) − 1 0 U T T = A G + ε ( A R + B G ) = A ^ G ^ .

Therefore, G ^ is the DCGI of A ^ , i.e., G ^ = A ^ #⃝ by Definition 3.1.□

Theorem 3.4

Let A ^ = A + ε B ∈ D n , n . Then, the DCGI A ^ #⃝ of A ^ exists if and only if Ind ( A ) = 1 and ( I n − A A + ) B ( I n − A A # ) = 0 .

Proof

Suppose that A is core invertible. Let the decomposition of A be as in (2.9), and the form of B be as in (2.15).

“ ⇒ ” Let the DCGI A ^ #⃝ of A ^ exists, then we have the index of A as 1, and B 4 = B 3 K − 1 L from Theorem 3.3. Thus, we obtain equation (3.19). Substituting equations (2.9), (2.11), (2.14), and (3.19) into ( I n − A A + ) B ( I n − A A # ) , we obtain

(3.21) ( I n − A A + ) B ( I n − A A # ) = U 0 0 0 I n − r B 1 B 2 B 3 B 3 K − 1 L 0 − K − 1 L 0 I n − r U T = U 0 0 B 3 B 3 K − 1 L 0 − K − 1 L 0 I n − r U T = U 0 0 0 − B 3 K − 1 L + B 3 K − 1 L U T = 0 .

“ ⇐ ” Let ( I n − A A + ) B ( I n − A A # ) = 0 . Since A # exists, the index of A is 1. Substituting equations (2.9), (2.11), (2.14), and (2.15) into ( I n − A A + ) B ( I n − A A # ) , we obtain

(3.22) ( I n − A A + ) B ( I n − A A # ) = U 0 0 0 I n − r B 1 B 2 B 3 B 4 0 − K − 1 L 0 I n − r U T = U 0 0 B 3 B 4 0 − K − 1 L 0 I n − r U T = U 0 0 0 − B 3 K − 1 L + B 4 U T .

It follows from ( I n − A A + ) B ( I n − A A # ) = 0 that B 4 = B 3 K − 1 L . Then, there is equation (3.20) from Theorem 3.3. According to Definition 3.1, DCGI exists.□

Theorem 3.5

Let A ^ = A + ε B ∈ D n , n , then the DCGI A ^ #⃝ of A ^ exists if and only if Ind ( A ) = 1 and ( I n − A A #⃝ ) B ( I n − A #⃝ A ) = 0 .

Proof

In the real field, it is known that A # A = A A # = A #⃝ A and A A + = A A #⃝ . According to Theorem 3.4, the DCGI A ^ #⃝ of A ^ exists if and only if Ind ( A ) = 1 and ( I n − A A #⃝ ) B ( I n − A #⃝ A ) = 0 .□

Next, we further discuss the characterizations of the existence of DCGI.

Theorem 3.6

Let A ^ = A + ε B ∈ D n , n , then the following conditions are equivalent:

  1. The DCGI of A ^ exists;

  2. The index of A is equal to 1, and B A # A + A X A + A A # B = B is consistent;

  3. The index of A is equal to 1, and B A + A + A X A + A A + B = B is consistent;

  4. The index of A is equal to 1, and B A #⃝ A + A X A + A A #⃝ B = B is consistent.

Proof

From Theorems 2.6, 2.7, and 3.3, we know that the existence of DCGI is equivalent to the existence of DGGI; the existence of DGGI is equivalent to the existence of DMPGI with Ind ( A ) = 1 . Therefore, condition (1) indicates that DCGI exists or DGGI exists or DMPGI exists with Ind ( A ) = 1 .

“ ( 1 ) ⇒ ( 2 ) ” Let DCGI exists, then A ^ # = A # + ε P of A ^ = A + ε B exists, so the index of A is 1 and A ^ A ^ # A ^ = A ^ , i.e., B A # A + A P A + A A # B = B . Therefore, B A # A + A X A + A A # B = B is consistent.

“ ( 1 ) ⇐ ( 2 ) ” Let the index of A be 1 and B A # A + A X A + A A # B = B be consistent. By applying equations (2.9), (2.10), (2.14), and (2.15) into B A # A + A X A + A A # B = B , we obtain

(3.23) U B 1 B 2 B 3 B 4 K − 1 Σ − 1 K − 1 Σ − 1 K − 1 L 0 0 Σ K Σ L 0 0 U T + U Σ K Σ L 0 0 X 1 X 2 X 3 X 4 Σ K Σ L 0 0 U T + U Σ K Σ L 0 0 K − 1 Σ − 1 K − 1 Σ − 1 K − 1 L 0 0 B 1 B 2 B 3 B 4 U T = U B 1 B 2 B 3 B 4 U T ,

where X = U X 1 X 2 X 3 X 4 U T , B 1 is an r -square matrix, and r = rank ( A ) .

By simplifying equation (3.23), we obtain

U B 1 B 2 B 3 B 4 U T = U B 1 + Σ K X 1 Σ K + Σ L X 3 Σ K + B 1 + K − 1 L B 3 B 3 B 1 K − 1 L + Σ K X 1 Σ L + Σ L X 3 Σ L + B 2 + K − 1 L B 4 B 3 K − 1 L U T .

Therefore, B 4 = B 3 K − 1 L .

To sum up, if Ind ( A ) = 1 and B 4 = B 3 K − 1 L , then the dual index of A ^ is 1 from Theorem 2.2. By Theorem 3.3, we obtain that the DCGI of A ^ exists.

Similarly, conditions (1) and (3) are equivalent, and conditions (1) and (4) are equivalent.□

3.3 Compact formula for DCGI

The following is a compact formula for DCGI.

Theorem 3.7

Let A ^ = A + ε B ∈ D n , n and the DCGI of A ^ exists, then

(3.24) A ^ #⃝ = A ^ # A ^ A ^ +

(3.25) = A #⃝ + ε ( − A #⃝ B A + + A # B A + − A # B A #⃝ + A #⃝ ( B A + ) T ( I n − A A + ) + ( I n − A A # ) B A # A #⃝ ) .

Proof

According to Theorems 2.6, 2.7, and 3.3, if A ^ #⃝ exists, then A ^ # and A ^ + exist. Write X ^ = A ^ # A ^ A ^ + . It is easy to check that

(3.26) A ^ X ^ A ^ = A ^ A ^ # A ^ A ^ + A ^ = A ^ A ^ + A ^ = A ^ ,

(3.27) A ^ X ^ 2 = A ^ A ^ # A ^ A ^ + A ^ # A ^ A ^ + = A ^ A ^ + A ^ A ^ # A ^ + = A ^ A ^ # A ^ + = A ^ # A ^ A ^ + = X ^ .

Since ( A ^ X ^ ) T = ( A ^ A ^ # A ^ A ^ + ) T = ( A ^ A ^ + ) T = A ^ A ^ + and A ^ X ^ = A ^ A ^ # A ^ A ^ + = A ^ A ^ + , we obtain

(3.28) ( A ^ X ^ ) T = A ^ X ^ .

To sum up, we obtain A ^ #⃝ = X ^ = A ^ # A ^ A ^ + .

Substituting equations (1.2) and (1.3) into equation (3.24) gives (3.25).□

In addition, substituting equations (2.9), (2.10), (2.11), (2.13), (2.14), and (3.19) into equation (3.25), we obtain the following Theorem 3.8.

Theorem 3.8

Let the DCGI of A ^ = A + ε B ∈ D n , n exists, A and B be as forms in equations (2.9) and (2.15), respectively. Then,

(3.29) A ^ #⃝ = U ( Σ K ) − 1 0 0 0 U T + ε U − K − 1 L B 3 ( Σ K ) − 2 − ( Σ K ) − 1 B 1 ( Σ K ) − 1 K − 1 Σ − 2 ( B 3 K − 1 ) T B 3 ( Σ K ) − 2 0 U T .

3.4 Relationships among some dual generalized inverses

The MPDGI is a very interesting inverse, which is useful for solving different kinematic problems [4]. Obviously, when the real part of a dual matrix is a nonsingular matrix, its MPDGI is equal to its DMPGI. Similarly, we consider the laws and properties of DCGI in the form of A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ and DGGI in the form of A ^ # = A # − ε A # B A # , as well as relationships among those dual generalized inverses.

Theorem 3.9

Let the DCGI A ^ #⃝ of A ^ = A + ε B ∈ D n , n exists, where rank ( A ) = r . Then A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ if and only if Ind ( A ) = 1 and

(3.30) ( I n − A A + ) B = 0 .

Proof

Suppose that A is core invertible. Let the decomposition of A be as in equation (2.9), and the form of B be as in equation (2.15). Then,

(3.31) − A #⃝ B A #⃝ = U − ( Σ K ) − 1 B 1 ( Σ K ) − 1 0 0 0 U T ,

(3.32) − A #⃝ B A + + A # B A + = U ( Σ K ) − 1 K − 1 L ( B 3 K T + B 4 L T ) Σ − 1 0 0 0 U T ,

(3.33) − A # B A #⃝ = U − ( Σ K ) − 1 B 1 ( Σ K ) − 1 − ( Σ K ) − 1 K − 1 L B 3 ( Σ K ) − 1 0 0 0 U T ,

(3.34) A #⃝ ( B A + ) T ( I n − A A + ) = U 0 ( Σ K ) − 1 ( B 3 K T Σ − 1 + B 4 L T Σ − 1 ) T 0 0 U T ,

(3.35) ( I n − A A # ) B A # A #⃝ = U − K − 1 L B 3 ( Σ K ) − 2 0 B 3 ( Σ K ) − 2 0 U T ,

(3.36) ( I n − A A + ) B = U 0 0 0 I n − r B 1 B 2 B 3 B 4 U T = U 0 0 B 3 B 4 U T .

“ ⇒ ” Let the DCGI A ^ #⃝ of A ^ = A + ε B ∈ D n , n exists and A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ . According to Theorem 3.3, we obtain Ind ( A ) = 1 and B 4 = B 3 K − 1 L . To make A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ , from the compact formula for DCGI (3.25), we obtain

A #⃝ B A #⃝ = − A #⃝ B A + + A # B A + − A # B A #⃝ + A #⃝ ( B A + ) T ( I n − A A + ) + ( I n − A A # ) B A # A #⃝ .

It follows from (3.31)–(3.35) that B 3 ( Σ K ) − 2 = 0 , so B 3 = 0 . Since B 4 = B 3 K − 1 L , then B 4 = 0 . By applying equation (3.36), B 3 = 0 and B 4 = 0 , we derive ( I n − A A + ) B = 0 . Thus, Ind ( A ) = 1 and ( I n − A A + ) B = 0 are proved.

“ ⇐ ” Let Ind ( A ) = 1 and ( I n − A A + ) B = 0 , then ( I n − A A + ) B ( I n − A A # ) = 0 . According to Theorem 2.4, the DCGI of A ^ exists. We know that the real part of the DCGI is A #⃝ , i.e., A ^ #⃝ = A #⃝ + ε R . Put equations (2.9), (2.11), and (2.15) into equation (3.36) to obtain ( I n − A A + ) B = 0 . From ( I n − A A + ) B = 0 , we have B 3 = 0 and B 4 = 0 . Put B 3 = 0 and B 4 = 0 into equation (3.25) to obtain

(3.37) R = − A #⃝ B A + + A # B A + − A # B A #⃝ + A #⃝ ( B A + ) T ( I n − A A + ) + ( I n − A A # ) B A # A #⃝ = U − ( Σ K ) − 1 B 1 ( Σ K ) − 1 0 0 0 U T = − A #⃝ B A #⃝ .

Therefore, A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ .□

Since it is well known that ( I n − A A + ) B = 0 if and only if rank A B = rank ( A ) , we obtain the following Theorem 3.10.

Theorem 3.10

Let the DCGI A ^ #⃝ of A ^ = A + ε B ∈ D n , n exists, where rank ( A ) = r . Then, A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ is equivalent to Ind ( A ) = 1 and rank A B = rank ( A ) .

Next, we continue to analyze DGGI in the form of A ^ # = A # − ε A # B A # .

Theorem 3.11

Let the DGGI A ^ # of A ^ = A + ε B ∈ D n , n exists, where rank ( A ) = r . Then, A ^ # = A # − ε A # B A # is equivalent to

(3.38) B ( I n − A A # ) = 0 a n d ( I n − A A # ) B = 0 .

Proof

Suppose that A is core invertible. Let the decomposition of A be as in equation (2.9), the decomposition of A # be as in equation (2.14), and the form of B be as in equation (2.15). Then, we have

( A # ) 2 B ( I n − A A # ) = U 0 ( Σ K ) − 2 ( − B 1 K − 1 L + B 2 − K − 1 L B 3 K − 1 L + K − 1 L B 4 ) 0 0 U T

and

(3.39) ( I n − A A # ) B ( A # ) 2 = U − K − 1 L B 3 ( Σ K ) − 2 − K − 1 L B 3 ( Σ K ) − 2 K − 1 L B 3 ( Σ K ) − 2 B 3 ( Σ K ) − 2 K − 1 L U T .

“ ⇒ ” Let the DGGI A ^ # of A ^ ∈ D n , n exists, then Ind ( A ^ ) = 1 from Theorem 3.3. Therefore, Ind ( A ) = 1 and B 4 = B 3 K − 1 L from Theorem 2.2. Thus,

(3.40) ( A # ) 2 B ( I n − A A # ) = U 0 − ( Σ K ) − 2 B 1 K − 1 L + ( Σ K ) − 2 B 2 0 0 U T .

If A ^ # = A # − ε A # B A # , by equations (1.3), (3.39), and (3.40), we have ( A # ) 2 B ( I n − A A # ) = 0 and ( I n − A A # ) B ( A # ) 2 = 0 , so B 2 = B 1 K − 1 L and B 3 = 0 . It follows that

(3.41) B = U B 1 B 1 K − 1 L 0 0 U T .

Therefore, we have

B ( I n − A A # ) = U B 1 B 1 K − 1 L 0 0 U T U 0 − K − 1 L 0 I n − r U T = 0 , ( I n − A A # ) B = U 0 − K − 1 L 0 I n − r U T U B 1 B 1 K − 1 L 0 0 U T = 0 ,

i.e., (3.38) is established.

“ ⇐ ” Assuming B ( I n − A A # ) = 0 and ( I n − A A # ) B = 0 , it is easy to check that ( I n − A A # ) B ( I n − A # A ) = 0 , ( A # ) 2 B ( I n − A A # ) = 0 , and ( I n − A A # ) B ( A # ) 2 = 0 . According to Lemma 1.3, the DGGI of A ^ exists. Therefore, by using (1.3), we obtain A ^ # = A # − ε A # B A # .□

In Theorems 3.12, 3.13, and 3.14, we consider the relationships among A ^ # = A # − ε A # B A # , A ^ + = A + − ε A + B A + , and A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ .

Theorem 3.12

Let A ^ = A + ε B ∈ D n , n , then DGGI exists and A ^ # = A # − ε A # B A # if and only if  Ind ( A ) = 1 , DMPGI exists, and A ^ + = A + − ε A + B A + .

Proof

Suppose that A is group invertible. Let the decomposition of A be as in equation (2.9) and the form of B be as in equation (2.15).

“ ⇒ ” Let DGGI exists and A ^ # = A # − ε A # B A # , then B ( I n − A A # ) = 0 and ( I n − A A # ) B = 0 from Theorem 3.11.

Substituting equations (2.9), (2.14), and (2.15) into ( I n − A A # ) B = 0 , we obtain

( I n − A A # ) B = U 0 − K − 1 L 0 I n − r B 1 B 2 B 3 B 4 U T = U − K − 1 L B 3 − K − 1 L B 4 B 3 B 4 U T = 0 .

Therefore, B 3 = 0 and B 4 = 0 . Substituting equations (2.9) and (2.14) into B ( I n − A A # ) = 0 , it follows that

B ( I n − A A # ) = U B 1 B 2 0 0 0 − K − 1 L 0 I n − r U T = U 0 − B 1 K − 1 L + B 2 0 0 U T = 0 ,

i.e., B 2 = B 1 K − 1 L .

Since B 3 = 0 , B 4 = 0 , and B 2 = B 1 K − 1 L , we have

(3.42) B = U B 1 B 1 K − 1 L 0 0 U T .

Since K K T + L L T = I r , applying equations (2.16), (2.17), and (3.42), we have

B ( I n − A + A ) = U B 1 B 1 K − 1 L 0 0 U T U I r − K T K − K T L − L T K I n − r − L T L U T = U B 1 − B 1 K T K − B 1 K − 1 ( I r − K K T ) K − B 1 K T L + B 1 K − 1 L − B 1 K − 1 ( I r − K K T ) L 0 0 U T = 0

and

( I n − A A + ) B = U 0 0 0 I n − r U T U B 1 B 1 K − 1 L 0 0 U T = 0 .

By applying Lemma 1.2, we can conclude that DMPGI exists and A ^ + = A + − ε A + B A + .

Because the DGGI exists, Ind ( A ) = 1 . To sum up, Ind ( A ) = 1 , DMPGI exists, and A ^ + = A + − ε A + B A + .

“ ⇐ ” Let Ind ( A ) = 1 , DMPGI exists, and A ^ + = A + − ε A + B A + . By applying Lemma 1.2, we know ( I n − A A + ) B = 0 and B ( I n − A + A ) = 0 . Substituting equations (2.9), (2.11), and (2.15) into ( I n − A A + ) B = 0 , we obtain

( I n − A A + ) B = U 0 0 0 I n − r B 1 B 2 B 3 B 4 U T = U 0 0 B 3 B 4 U T = 0 ,

i.e., B 3 = 0 and B 4 = 0 .

Substituting equations (2.9), (2.11), and (2.15) into B ( I n − A + A ) = 0 and by applying B 3 = 0 and B 4 = 0 , we obtain

B ( I n − A + A ) = U B 1 B 2 0 0 I r − K T K − K T L − L T K I n − r − L T L U T = U B 1 − B 1 K T K − B 2 L T K − B 1 K T L + B 2 − B 2 L T L 0 0 U T = 0 .

Therefore,

B 1 = B 1 K T K + B 2 L T K = ( B 1 K T + B 2 L T ) K , (3.43a) B 2 = B 2 L T L + B 1 K T L = ( B 2 L T + B 1 K T ) L . (3.43b)

Since Ind ( A ) = 1 , K is nonsingular. Applying (3.43a) gives

(3.44) B 1 K T + B 2 L T = B 1 K − 1 .

Substituting equation (3.44) into equation (3.43b), we have B 2 = B 1 K − 1 L .

Since B 3 = 0 , B 4 = 0 , and B 2 = B 1 K − 1 L and by substituting equations (2.9) and (2.11) into B ( I n − A A # ) and ( I n − A A # ) B , we have B ( I n − A A # ) = 0 and ( I n − A A # ) B = 0 . According to Theorem 3.11, DGGI exists and A ^ # = A # − ε A # B A # .□

Theorem 3.13

Let A ^ = A + ε B ∈ D n , n . If Ind ( A ) = 1 , DMPGI exists and A ^ + = A + − ε A + B A + , then DCGI exists and A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ .

Proof

Since Ind ( A ) = 1 , DMPGI exists and A ^ + = A + − ε A + B A + , then ( I n − A A + ) B = 0 by applying Lemma 1.2.

According to Theorem 3.9, if Ind ( A ) = 1 and ( I n − A A + ) B = 0 , then the DCGI A ^ #⃝ of A ^ exists, and A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ . □

Theorem 3.14

Let A ^ = A + ε B ∈ D n , n . If A ^ # exists, and A ^ # = A # − ε A # B A # , then DCGI exists and A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ .

Proof

If A ^ # exists, and A ^ # = A # − ε A # B A # , then Ind ( A ) = 1 , DMPGI exists and A ^ + = A + − ε A + B A + from Theorem 3.12. Then, DCGI exists and A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ from Theorem 3.13.□

Example 3.2

Let A ^ = 1 0 0 0 + ε 0 1 0 0 . By applying equations (1.2), (1.3), and (3.25), we have

A ^ + = 1 0 0 0 + ε 0 0 1 0 , A ^ # = 1 0 0 0 + ε 0 1 0 0 , A ^ #⃝ = 1 0 0 0 + ε 0 0 0 0 = 1 0 0 0

and

A #⃝ − ε A #⃝ B A #⃝ = 1 0 0 0 , A ^ P = A + − ε A + B A + = 1 0 0 0 , A # − ε A # B A # = 1 0 0 0 .

It is easy to see that A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ , A ^ + ≠ A ^ P , and A ^ # ≠ A # − ε A # B A # . This means that when DCGI exists and A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ , there is not necessarily A ^ + = A ^ P = A + − ε A + B A + or A ^ # = A # − ε A # B A # .

3.5 Symmetric dual matrix

We know that in real field, the index of a symmetric matrix must be equal to 1, and its core inverse is equal to its Moore-Penrose inverse and its group inverse. But it is not true for some symmetric dual matrices. Even some symmetric dual matrices do not have DCGIs and DGGIs. For example,

Example 3.3

A ^ = A + ε B = a 0 0 0 b 0 0 0 0 + ε 0 0 0 0 0 0 0 0 c , where a , b , and c are not 0. Because rank ( A ) = rank ( A 2 ) = 2 , rank A B ( I 3 − A A # ) = 3 , and 2 ≠ 3 , we can obtain that the dual index of A ^ is not 1, A ^ does not satisfy Theorems 2.6 and 3.3. Therefore, A ^ has no DGGI and DCGI, as well as A ^ no DMPGI from Theorem 2.8.

Theorem 3.15

If the dual matrix A ^ is a symmetric dual matrix, and the dual index of A ^ is 1, then

(3.45) A ^ # = A ^ + = A ^ #⃝ .

Proof

Let A ^ = A + ε B ∈ D n , n , A ^ T = A ^ , rank ( A ) = r , and the dual index of A ^ be 1. According to Theorems 2.6, 2.7, and 3.3, the DGGI, DMPGI, and DCGI of A ^ exist simultaneously. Since A ^ = A ^ T , we obtain that A is symmetrical, B is symmetrical, A 2 is symmetrical, and ( A 2 ) # = ( A 2 ) + = ( A A T ) + = ( A T A ) + . Therefore, we have

A + = A # , A + B A + = A # B A # ( A T A ) + B T ( I n − A A + ) = ( A # ) 2 B ( I n − A # A ) , ( I n − A + A ) B T ( A A T ) + = ( I n − A A # ) B ( A # ) 2 .

Then, by applying equations (1.2) and (1.3), we derive that A ^ # = A ^ + .

It follows from A ^ # = A ^ + and A ^ #⃝ = A ^ # A ^ A ^ + in equation (3.24) that A ^ #⃝ = A ^ # A ^ A ^ + = A ^ # A ^ A ^ # = A ^ # = A ^ + . Therefore, A ^ # = A ^ + = A ^ #⃝ .□

Theorem 3.16

If the DCGI A ^ #⃝ = A #⃝ + ε R of A ^ = A + ε B ∈ D n , n exists, and A ^ is a symmetric dual matrix, then

(3.46) A ^ #⃝ = ( A ^ #⃝ ) T = A #⃝ + ε ( − A #⃝ B A #⃝ + ( A #⃝ ) 2 B ( I n − A A #⃝ ) + ( I n − A A #⃝ ) B ( A #⃝ ) 2 ) .

Proof

According to Theorems 2.6, 2.7, and 3.3, if the DCGI of A ^ exists, then DGGI and DMPGI exist.

Then A ^ #⃝ = A ^ # A ^ A ^ + = A ^ + A ^ A ^ + = A ^ + by equation (3.45). From ( A ^ + ) T = A ^ + (see [23]), we have ( A ^ #⃝ ) T = A ^ #⃝ .

Since A ^ is a symmetric dual matrix, we have A T = A and B T = B . Moreover, A #⃝ = A # = A + = ( A #⃝ ) T = ( A # ) T = ( A + ) T .

According to equation (3.25), we can obtain

A ^ #⃝ = A #⃝ + ε ( − A #⃝ B A + + A # B A + − A # B A #⃝ + A #⃝ ( B A + ) T ( I n − A A + ) + ( I n − A A # ) B A # A #⃝ ) = A #⃝ + ε ( − A #⃝ B A #⃝ + A #⃝ ( B A + ) T ( I n − A A + ) + ( I n − A A # ) B A # A #⃝ ) = A #⃝ + ε ( − A #⃝ B A #⃝ + A #⃝ ( B A #⃝ ) T ( I n − A A #⃝ ) + ( I n − A A #⃝ ) B A #⃝ A #⃝ ) = A #⃝ + ε ( − A #⃝ B A #⃝ + A #⃝ ( A #⃝ ) T B T ( I n − A A #⃝ ) + ( I n − A A #⃝ ) B A #⃝ A #⃝ ) = A #⃝ + ε ( − A #⃝ B A #⃝ + ( A #⃝ ) 2 B ( I n − A A #⃝ ) + ( I n − A A #⃝ ) B ( A #⃝ ) 2 ) ,

i.e., (3.46).□

Next, we consider some properties of DCGI, DGGI, and DMPGI in special forms when the research object is a symmetric dual matrix.

Lemma 3.17

[34] Let A ∈ R n , n and B ∈ R n , n . Then, ℛ ( B ) ⊆ ℛ ( A ) if and only if rank A B = rank ( A ) .

Theorem 3.18

Let A ^ = A + ε B be an n-order symmetric dual matrix. Then, the following conditions are equivalent:

  1. The DCGI A ^ #⃝ of A ^ exists, and A ^ #⃝ = A ^ # = A ^ + = A ^ P = A + − ε A + B A + ;

  2. ( I n − A A + ) B = 0 or ( I n − A A # ) B = 0 or ( I n − A A #⃝ ) B = 0 ;

  3. rank A B = rank ( A ) ;

  4. ℛ ( B ) ⊆ ℛ ( A ) ;

  5. B ( I n − A A + ) = 0 or B ( I n − A A # ) = 0 or B ( I n − A A #⃝ ) = 0 .

Proof

Let A ^ ∈ D n , n , rank ( A ) = r , and A ^ T = A ^ . Since A ^ is a symmetric dual matrix, then the real part A is symmetric, and A #⃝ = A + = A # . Therefore, condition (1) is A ^ #⃝ = A ^ # = A ^ + = A ^ P = A + − ε A + B A + = A # − ε A # B A # = A #⃝ − ε A #⃝ B A #⃝ , and conditions (2) and (5) are equivalent.

“ ( 1 ) ⇒ ( 2 ) ” Let A ^ #⃝ = A ^ # = A ^ + = A ^ P = A # − ε A # B A # . From Theorem 3.11, we obtain B ( I n − A A # ) = 0 and ( I n − A A # ) B = 0 . Therefore, we obtain ( I n − A A # ) B = 0 . Since A #⃝ = A + = A # , condition (2) is established.

“ ( 1 ) ⇐ = ( 2 ) ” Let ( I n − A A # ) B = 0 . Since A ^ is symmetrical, B ( I n − A A # ) = 0 . Applying Theorem 3.11 gives that DGGI A ^ # of A ^ = A + ε B exists and A ^ # = A # − ε A # B A # .

According to Theorems 2.6, 2.7, and 3.3, when A ^ # exists, A ^ #⃝ and A ^ + exist, and the dual index of A ^ is 1. According to Theorem 3.15, A ^ #⃝ = A ^ + = A ^ # . Then A ^ #⃝ = A ^ + = A ^ # = A # − ε A # B A # . Considering that A ^ is a symmetric dual matrix, then A #⃝ = A + = A # . Therefore, equation (1) is established.

According to Theorem 3.10, conditions (1) and (3) are equivalent. According to Lemma 3.17, conditions (3) and (4) are equivalent.□

According to Theorems 3.15 and 3.16, it is easy to obtain the following Corollary 3.19.

Corollary 3.19

Let A ^ = A + ε B ∈ D n , n be a symmetric dual matrix. If the DCGI, DGGI, and DMPGI of A ^ exist, then they are symmetrical and equal.

4 Applications of DCGI in linear dual equations

In this section, we use two examples to illustrate some applications of DCGI in solving linear dual equations.

First, we consider solving a consistent linear dual equation by DCGI in Example 4.1. We give a general solution to the consistent dual equation.

Example 4.1

Let A ^ x ^ = b ^ be a consistent equation, where

A ^ = A + ε B = 1 0 0 0 + ε 1 1 1 0 = 1 + ε ε ε 0 ,

b ^ = 1 0 + ε 1 1 = 1 + ε ε , and x ^ = x 1 ^ x 2 ^ , x i ^ = x i + ε x i ′ , i = 1 , 2 .

It is easy to check that rank ( A 2 ) = rank ( A ) = rank A B ( I n − A A # ) = 1 . By applying Theorem 2.1, we obtain that the dual index of A ^ is 1, i.e., the DCGI A ^ #⃝ exists. Applying (3.25) gives

(4.1) A ^ #⃝ = G + ε R = 1 0 0 0 + ε − 1 1 1 0 = 1 − ε ε ε 0 .

Thus,

A ^ #⃝ b ^ = 1 − ε ε ε 0 1 + ε ε = 1 ε .

Furthermore, let

(4.2) x ^ = A ^ #⃝ b ^ + ( I 2 − A ^ #⃝ A ^ ) w ^ = 1 ε + 0 − ε − ε 1 w ^ ,

where w ^ is an arbitrary n -by-1 dual column vector.

By substituting equation (4.2) into A ^ x ^ = b ^ , we can obtain

A ^ x ^ − b ^ = A ^ ( A ^ #⃝ b ^ + ( I 2 − A ^ #⃝ A ^ ) w ^ ) − b ^ = A ^ A ^ #⃝ b ^ − b ^ + ( A ^ − A ^ A ^ #⃝ A ^ ) w ^ = 1 + ε ε ε 0 1 ε − b ^ = 1 + ε ε − b ^ = 1 + ε ε − 1 + ε ε = 0 ,

i.e., (4.2) is the solution to A ^ x ^ = b ^ .

Meanwhile, let x ^ be any solution to A ^ x ^ = b ^ . Pre-multiplying A ^ x ^ = b ^ by A ^ #⃝ , we obtain A ^ #⃝ A ^ x ^ = A ^ #⃝ b ^ , and then

x ^ = A ^ #⃝ b ^ + x ^ − A ^ #⃝ A ^ x ^ = A ^ #⃝ b ^ + ( I 2 − A ^ #⃝ A ^ ) x ^ = 1 ε + 0 − ε − ε 1 x ^ .

Therefore, each solution to A ^ x ^ = b ^ can be written as equation (4.2) in which w ^ = x ^ .

To sum up, equation (4.2) is the general solution to A ^ x ^ = b ^ .

In order to solve inconsistent dual linear equation, Udwadia [23] introduced the norm of the dual vector. Consider the m -by-1 dual vector u i ^ = p i + ε q i . We write

(4.3) ‖ u i ^ ‖ 2 = ( p i + ε q i ) T ( p i + ε q i ) = ‖ p i ‖ 2 + 2 ε p i T q i ,

where p i ≠ 0 and ‖ p i ‖ 2 = p i T p i . By using the right-most expression in equation (4.3), one norm of the dual vector u i ^ is given as follows:

(4.4) ⟨ u i ^ ⟩ ≔ ‖ p i ‖ + ‖ q i ‖ .

In equation (4.3), the dual norm is used to determine the magnitude of the error. Udwadia [23] introduced the analog of the least squares solution of any inconsistent dual equation A ^ x ^ = b ^ and gives the corresponding solution – analog of the least squares solution x ^ = A ^ ( 1 , 3 ) b ^ + ( I n − A ^ ( 1 , 3 ) A ^ ) h ^ , where h ^ is an arbitrary dual column vector and A ^ ( 1 , 3 ) exists. It can be found that the real part x of the analog of the least squares solution is the least squares solution to equation A x = b , where A , b , and x are matched with the real parts of A ^ , b ^ , and x ^ , respectively.

Example 4.2

Let the inconsistent equation be A ^ x ^ = b ^ , where A ^ = A + ε B = 4 2 2 1 + ε 10 10 9 7 = 4 + 10 ε 2 + 10 ε 2 + 9 ε 1 + 7 ε , b ^ = 0 1 + ε 1 0 = ε 1 , and x ^ = x 1 ^ x 2 ^ , x i ^ = x i + ε x i ′ , i = 1 , 2 , i.e.,

(4.5) 4 + 10 ε 2 + 10 ε 2 + 9 ε 1 + 7 ε x 1 ^ x 2 ^ = ε 1 .

Then,

A ^ + = 0.1600 − 0.6880 ε 0.0800 − 0.1840 ε 0.0800 − 0.1440 ε 0.0400 + 0.0080 ε .

By applying the Result 12 of Udwadia in [23], the analog of the least squares solution of the inconsistent equation is

x ^ = A ^ + b ^ + ( I 2 − A ^ + A ^ ) h ^ = 0.0800 − 0.0240 ε 0.0400 + 0.0880 ε + 0.2000 + 0.8000 ε − 0.4000 − 0.6000 ε − 0.4000 − 0.8000 ε 0.8000 − 0.8000 ε h ^ ,

where h ^ is an arbitrary n -by-1 dual vector. The norm of the error

(4.6) ⟨ e ∗ ^ ⟩ = ⟨ A ^ x ^ − b ^ ⟩ = ⟨ A ^ A ^ + b ^ − b ^ ⟩ = 1.9715 .

According to Theorem 2.1, we have rank ( A 2 ) = rank ( A ) = rank A B ( I n − A A # ) = 1 , i.e., the dual index of A ^ is 1. From Theorem 3.3, it can be seen that A ^ #⃝ exists. Then,

A ^ #⃝ = G + ε R = 0.1600 − 0.6720 ε 0.0800 − 0.1760 ε 0.0800 − 0.1760 ε 0.0400 − 0.0080 ε

by the compact formula (3.25). Denote

(4.7) x ^ = A ^ #⃝ b ^ + ( I − A ^ #⃝ A ^ ) w ^ = 0.0800 − 0.0160 ε 0.0400 + 0.0720 ε + 0.2000 + 0.7200 ε − 0.4000 − 0.6400 ε − 0.4000 − 0.4400 ε 0.8000 − 0.7200 ε w ^ ,

where w ^ is an arbitrary n -by-1 dual vector. The norm of the error is

(4.8) ⟨ e ^ ⟩ = ⟨ A ^ x ^ − b ^ ⟩ = ⟨ A ^ A ^ #⃝ b ^ − b ^ ⟩ = ⟨ u 2 ^ ⟩ = ∥ m 2 ∥ + ∥ n 2 ∥ = 0.4000 − 0.8000 + 0.2800 1.0400 = 1.9715 .

Therefore, from equations (4.6) and (4.8), we see that ⟨ e ∗ ^ ⟩ = ⟨ e ^ ⟩ = 1.9715 , i.e., equation (4.7) is also the analog of the least-squares solution of equation (4.5).

The two examples in this section calculate the DCGIs of the two dual matrices through the compact formula (3.25). When the dual index of any dual matrix is 1, its DCGI exists. On this basis, we can obtain DCGI directly through the compact formula (3.25). However, in order to reduce the amount of calculation, we can first consider equation (3.30) in Theorem 3.9. If the dual matrix satisfies equation (3.30), then A ^ #⃝ = A #⃝ − ε A #⃝ B A #⃝ . Otherwise, we have to use the compact formula (3.25).

5 Conclusion

The first part of this article provides some new findings about dual index 1 of dual matrices, including the characterizations of the dual index 1. We obtain that DGGI exists if and only if the dual index is 1. Furthermore, when the dual index is 1, DMPGI exists and the real part index of the dual Moore-Penrose generalized invertible matrix is 1, and vice versa. The second part of this article explores DCGI systematically. Some results from the second part of the article are as follows:

  1. If a DCGI of a dual matrix exists, it is unique.

  2. If DCGI exists, a compact formula for DCGI is given.

  3. We provide a series of equivalent characterizations of the existence of DCGI, e.g., the dual index is 1 if and only if DCGI exists.

  4. Relations among MPDGI, DMPGI, DCGI, and DGGI are proved.

In the third part, DCGI is applied to linear dual equations through a consistent dual equation and an inconsistent dual equation.

Acknowledgement

The authors wish to extend their sincere gratitude to the referees for their precious comments and suggestions.

  1. Funding information: This work was supported partially by the Research Fund Project of Guangxi University for Nationalities [grant number 2019KJQD03], the National Natural Science Foundation of China [grant number 12061015], the Thousands of Young and Middle-aged Key Teachers Training Programme in Guangxi Colleges and Universities [grant number GUIJIAOSHIFAN2019-81HAO], and the Innovation Project of Guangxi Graduate Education [grant number YCSW2022243].

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state that there is no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2023-01-04
Revised: 2023-05-17
Accepted: 2023-05-19
Published Online: 2023-07-06

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
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  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
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  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
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  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
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  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
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  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
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  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
https://doi.org/10.1515/math-2022-0592
Keywords for this article
dual core generalized inverse; dual index; dual Moore-Penrose generalized inverse; dual group generalized inverse; Moore-Penrose dual generalized inverse; dual analog of least-squares solutions
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  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
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  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
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  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
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  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
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  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
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  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
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  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
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  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
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  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
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  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
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  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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