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Let Cm×n and Z+ denote the set of all m×n complex matrices and the set of all positive integers, respectively. The symbols r(A) and Ind(A) stand for the rank and the index of A∈Cn×n, respectively. For a matrix A∈Cn×n, we assume that A0=In. Let Cn×nk be the set of all n×n complex matrices with index k. By CCMn we denote the set of all core matrices (or group invertible matrices), i.e.,
CCMn={A|A∈Cn×n,r(A)=r(A2)}. |
The Drazin inverse [1] of A∈Cn×nk, denoted by AD, is the unique matrix X∈Cn×n satisfying:
XAk+1=Ak,XAX=X and AX=XA. | (1.1) |
Especially, when A∈CCMn, then X that satisfies (1.1) is called the group inverse of A and is denoted by A#. The Drazin inverse has been widely applied in different fields of mathematics and its applications. Here we will mention only some of them. The perturbation theory and additive results for the Drazin inverse were investigated in [2,3,4,5]. In [6], the algorithms for the computation of the Drazin inverse of a polynomial matrix are presented based on the discrete Fourier transformation. Karampetakis and Stanimiroviˊc [7] presented two algorithms for symbolic computation of the Drazin inverse of a given square one-variable polynomial matrix, which was effective with respect to CPU time and the elimination of redundant computations. Some representations of the W-weighted Drazin inverse were investigated and the computational complexities of the representations were also estimated in [8]. Kyrchei [9] generalized the weighted Drazin inverse, the weighted DMP-inverse, and the weighted dual DMP-inverse [10,11,12] for the matrices over the quaternion skew field and provided their determinantal representations by using noncommutative column and row determinants. In [13], the authors considered the quaternion two-sided restricted matrix equations and gave their unique solutions by the DMP-inverse and dual DMP-inverse. For interesting properties of different kinds of generalized inverses see [14].
In 2018, Wang [15] introduced the weak group inverse of complex square matrices using the core-EP decomposition [16] and gave its certain characterizations.
Definition 1.1. Let A∈Cn×nk. Then the unique solution of the system
![]() |
is the weak group inverse of A denoted by AⓌ.
Recently, there has been a huge interest in the weak group inverse. For example, Wang et al. [17] compared the weak group inverse with the group inverse of a matrix. In [18], the weak group inverse was introduced in *-rings and characterized by three equations (see also [19,20]). The weak group inverse in the setting of rectangular matrices was considered in [21]. In 2021, Zhou and Chen [19] introduced the m-weak group inverse in the ring and presented its different characterizations.
Definition 1.2. Let R be a unitary ring with involution, a∈R and m∈Z+. If there exist x∈R and k∈Z+ such that
xak+1=ak, ax2=x, (ak)∗am+1x=(am)∗ak, |
then x is called the m-weak group inverse of a and in this case, a is m-weak group invertible.
In general, the m-weak group inverse of a may not be unique. If the m-weak group inverse of a is unique, then it is denoted by aⓌm.
In [22], we can find a relation between the weak core inverse and the m-weak group inverse as well as certain necessary and sufficient conditions that the Drazin inverse coincides with the m-weak group inverse of a complex matrix. It is interesting to note that X which satisfies (1.1) coincides with the m-weak group inverse on complex matrices, in which case X exists for every A∈Cn×n and is unique.
Now, we consider the system of equations
![]() |
(1.2) |
Motivated by the above discussion, we introduce a new characterization of the m-weak group inverse related with (1.2) and proved the existence and uniqueness of a solution of (1.2), for every A∈Cn×n. Some new characterizations of the m-weak group inverse are derived in terms of the range space, null space, rank equalities, and projectors. We present some representations of the m-weak group inverse involving some known generalized inverses and limit expressions as well as certain relations between the m-weak group inverse and other generalized inverses. Finally, we consider a relation between the m-weak group inverse and the nonsingular bordered matrix, which is applied to the Cramer's rule for the solution of the restricted matrix equation.
The paper is organized as follows: In Section 2, we present some well-known definitions and lemmas. In Section 3, we provide a new characterization, as well as certain representations and properties of the m-weak group inverse of a complex matrix. In Section 4, we provide several expressions of the m-weak group inverse which are useful in computation. In Section 5, we present some properties of the m-weak group inverse as well as the relationships between the m-weak group inverse and other generalized inverses by core-EP decomposition. In Section 6, we show the applications of the m-weak group inverse concerned with the bordered matrices and the Cramer's rule for the solution of the restricted matrix equation.
The symbols R(A), N(A) and A∗ denote the range space, null space and conjugate transpose of A∈Cm×n, respectively. The symbol In denotes the identity matrix of order n. Let PL,M be the projector on the space L along the M, where L,M≤Cn and L⊕M=Cn. For A∈Cm×n, PA represents the orthogonal projection onto R(A), i.e., PA=PR(A)=AA†. The symbols CPn and CHn represent the subsets of Cn×n consisting of all idempotent and Hermitian matrices, respectively, i.e.,
CPn={A|A∈Cn×n,A2=A},CHn={A|A∈Cn×n,A=A∗}. |
Let A∈Cm×n. The MP-inverse A† of A is the unique matrix X∈Cn×m satisfying the following four Penrose equations (see [14,23,24]):
(1) AXA=A, (2) XAX=X, (3) (AX)∗=AX, (4) (XA)∗=XA. |
A matrix X∈Cn×m that satisfies condition (1) above is called an inner inverse of A and the set of all inner inverses of A is denoted by A{1}, while a matrix X∈Cn×m that satisfies condition (2) above is called an outer inverse of A. A matrix X∈Cn×m that satisfies both conditions (1) and (2) is called a reflexive g-inverse of A. If a matrix X∈Cn×m satisfies
X=XAX, R(X)=T and N(X)=S, |
where T and S are the subspaces of Cn and Cm respectively, then X is an outer inverse of A with prescribed range and null space and it is denoted by A(2)T,S. If A(2)T,S exists, then it is unique. The notion of the core inverse on the CCMn was proposed and was denoted by [25,26,27]. The core inverse of A∈Cn×nk is the unique matrix X∈Cn×n satisfying
AX=PA, R(X)⊆R(A). |
In addition, it was proved that
![]() |
The core-EP inverse of A∈Cn×nk, denoted by is given in [28,29,30]. The core-EP inverse of A∈Cn×nk is the unique matrix X∈Cn×n satisfying
XAX=X, R(X)=R(X∗)=R(Ak). |
Moreover, it was proved that
![]() |
The DMP-inverse of A∈Cn×nk, denoted by AD,† was introduced in [10,11]. The DMP-inverse of A∈Cn×nk is the unique matrix X∈Cn×n satisfying
XAX=X, XA=ADA AkX=AkA†. |
Moreover, it was shown that
AD,†=ADAA†. |
Also, the dual DMP-inverse of A was introduced in [10], as A†,D=A†AAD.
The (B,C)-inverse of A∈Cm×n, denoted by A(B,C) [31,32], is the unique matrix X∈Cn×m satisfying
XAB=B,CAX=C,R(X)=R(B) and N(X)=N(C), |
where B,C∈Cn×m.
To discuss further properties of the m-weak group inverse, several auxiliary lemmas will be given. The first lemma gives the core-EP decomposition of a matrix A∈Cn×nk which will be a very useful tool throughout this paper.
Lemma 2.1. [16] Let A∈Cn×nk. Then there exists a unitary matrix U∈Cn×n such that
A=A1+A2=U[TS0N]U∗, | (2.1) |
A1=U[TS00]U∗, A2=U[000N]U∗, | (2.2) |
where T∈Ct×t is nonsingular with t=r(T)=r(Ak) and N is nilpotent of index k. The representation (2.1) is called the core-EP decomposition of A, while A1 and A2 are the core part and nilpotent part of A, respectively.
Following the representation (2.1) of a matrix A∈Cn×nk, we have the following representations of certain generalized inverses (see [15,16,33]):
![]() |
(2.3) |
AⓌ=U[T−1T−2S00]U∗, | (2.4) |
AD=U[T−1(Tk+1)−1Tk00]U∗, | (2.5) |
where Tk=k−1∑j=0TjSNk−1−j.
By direct computations, we get that A∈CCMn is equivalent with N=0, in which case
A#=U[T−1T−2S00]U∗, | (2.6) |
and
![]() |
(2.7) |
Let A∈Cn×nk be of the form (2.1) and let m∈Z+. The notations below will be frequently used in this paper:
M=S(In−t−N†N),△=(TT∗+MS∗)−1,Tm=m−1∑j=0TjSNm−1−j. |
Lemma 2.2. [34,Lemma 6] Let A∈Cn×nk be of the form (2.1). Then
A†=U[T∗△T∗△SN†M∗△N†−M∗△SN†]U∗. | (2.8) |
From (2.8) and [16,Theorem 2.2], we get that
AA†=U[It00NN†]U∗, | (2.9) |
A†A=U[T∗△T−T∗△MM∗△TN†N+M∗△M]U∗, | (2.10) |
Ak=U[TkTk00]U∗, | (2.11) |
Am=U[TmTm0Nm]U∗, | (2.12) |
PAk=Ak(Ak)†=U[It000]U∗, | (2.13) |
where t=r(Ak).
Lemma 2.3. [29,35,36] Let A∈Cn×nk and let m∈Z+. Then
Lemma 2.4. Let A∈Cn×nk and let m∈Z+. Then
Proof. Assume that A∈Cn×nk is of the form (2.1). By (2.3), (2.12) and (2.13), it follows that
![]() |
In this section, using the core-EP decomposition of a matrix A∈Cn×nk we will give another definition of the m-weak group inverse. Furthermore, some properties of the m-weak group inverse will be derived.
Theorem 3.1. Let A∈Cn×nk be given by (2.1) and let X∈Cn×n and m∈Z+. The system of equations
![]() |
(3.1) |
is consistent and has a unique solution X given by
![]() |
(3.2) |
Proof. If m=1, then X coincides with AⓌ. Clearly, X is the unique solution of (3.1) according to the definition of the weak group inverse. If m≠1, by (3.1), Lemmas 2.3 (d) and 2.4, it follows that
![]() |
Thus, by (2.3) and (2.12), we have that
![]() |
Definition 3.2. Let A∈Cn×nk and m∈Z+. The m-weak group inverse of A, denoted by AⓌm, is the unique solution of the system (3.1).
Remark 3.3. The m-weak group inverse is in some sense a generalization of the weak group inverse and Drazin inverse. We have the following:
(a) If m=1, then 1-weak group inverse of A∈Cn×nk coincides with the weak group inverse of A;
(b) If m≥k, then m-weak group inverse of A∈Cn×nk coincides with the Drazin inverse of A.
In the following example, we will show that the m-weak group inverse is different from some known generalized inverses.
Example 3.4. Let A=[I3I30N], where N=[010001000]. It can be verified that Ind(A)=3. By computations, we can check the following:
![]() |
AD,†=[I3H300], A†,D=[H1H4I3−H1H2−H4], AⓌ=[I3I300], |
where H1=[1200010001], H2=[111011001], H3=[110010000], H4=[121212011000] and N†=[000100010].
It is clear that
Theorem 3.5. Let A∈Cn×nk be decomposed by A=A1+A2 as in (2.1) and let m∈Z+. Then
(a) AⓌm is an outer inverse of A;
(b) AⓌm is a reflexive g-inverse of A1.
Proof. (a) By Lemmas 2.3 (d), 2.4 and the definition of AⓌm, it follows that
![]() |
(b) By (2.2) and (3.2), we get that
A1AⓌmA1=U[TS00][T−1T−(m+1)Tm00][TS00]U∗=U[TS00]U∗=A1. |
From [16,Theorem 3.4], we get . By the fact that
and the statement (a) above, it follows that
![]() |
Hence AⓌm is a reflexive g-inverse of A1.
Theorem 3.6. Let A∈Cn×nk and m∈Z+. Then
(a) r(AⓌm)=r(Ak).
(b) R(AⓌm)=R(Ak), N(AⓌm)=N((Ak)∗Am).
(c) AⓌm=A(2)R(Ak),N((Ak)∗Am).
Proof. (a) Assume that A is given by (2.1). From (2.11) and (3.2), it is clear that r(AⓌm)=t=r(Ak).
(b) Since implies that
and since r(AⓌm)=r(Ak), we get R(AⓌm)=R(Ak). From
and
we get
If
we get that
Then
, and by r(AⓌm)=r((Ak)∗Am), it follows that
(c) It is a direct consequence from Theorems 3.5 (a) and 3.6 (b).
Theorem 3.7. Let A∈Cn×nk and m∈Z+. Then
(a) AAⓌm=PR(Ak),N((Ak)∗Am);
(b) AⓌmA=PR(Ak),N((Ak)∗Am+1).
Proof. (a) From Theorem 3.5 (a), it follows that AAⓌm∈CPn. By the definition of AⓌm and (3.2), it can be proved that and r(AAⓌm)=r(AⓌm)=r(Ak)=t. Hence R(AAⓌm)=R(Ak). Similarly, we get that N(AAⓌm)=N(AⓌm)=N((Ak)∗Am). Therefore, AAⓌm=PR(Ak),N((Ak)∗Am).
(b) The proof follows similarly as for the part (a).
In this part, we represent some characterizations of the m-weak group inverse in terms of the range space, null space, rank equalities, and projectors.
The next theorem gives several characterizations of AⓌm.
Theorem 4.1. Let A∈Cn×nk, X∈Cn×n and let m∈Z+. Then the following hold:
(a) X=AⓌm.
(c) R(X)=R(Ak), Am+1X=PAkAm.
(d) R(X)=R(Ak), (Ak)∗Am+1X=(Ak)∗Am.
Proof. (a)⇒(b): This follows directly by Theorem 3.6 (b) and the definition of AⓌm.
(b)⇒(c): Premultiplying by Am, and by Lemma 2.4, it follows that
![]() |
(c)⇒(d): Premultiplying Am+1X=PAkAm by (Ak)∗, it follows that
(Ak)∗Am+1X=(Ak)∗PAkAm=(Ak)∗Am. |
(d)⇒(a): Let A be of the form (2.1). By (2.11) and R(X)=R(Ak), we obtain that
X=U[X1X200]U∗, |
where X1∈Ct×t and X2∈Ct×(n−t). Thus (Ak)∗Am+1X=(Ak)∗Am implies that
U[(Tk)∗Tm+1X1(Tk)∗Tm+1X2(˜T)∗Tm+1X1(˜T)∗Tm+1X2]U∗=U[(Tk)∗Tm(Tk)∗Tm(˜T)∗Tm(˜T)∗Tm]U∗, |
i.e., X1=T−1 and X2=(Tm+1)−1Tm, which imply X=U[T−1(Tm+1)−1Tm00]U∗=AⓌm.
By Theorem 3.5, it is known that AⓌm is an outer inverse of A∈Cn×nk, i.e., AⓌmAAⓌm=AⓌm. Using this result, we obtain some characterizations of AⓌm.
Theorem 4.2. Let A∈Cn×nk, X∈Cn×n and let m∈N+. Then the following conditions are equivalent:
(a) X=AⓌm.
(b) XAX=X, R(X)=R(Ak), N(X)=N((Ak)∗Am).
(d) XAX=X, R(X)=R(Ak), (Am)∗Am+1X∈CHn.
Proof. (a)⇒(b): It is a direct consequence of Theorem 3.6 (c).
(b)⇒(c): By XAX=X and R(X)=R(Ak), it follows that
![]() |
and
![]() |
Since AX, , we have
and XAX=X, we obtain that XAk+1=Ak.
(c)⇒(d): We have that
![]() |
and by R(Ak)=R(XAk+1)⊆R(X), we get R(XA)=R(Ak). Since , it follows that
![]() |
(d)⇒(a): Assume that A is of the form (2.1). From XAX=X and R(X)=R(Ak), we get that XAk+1=Ak. Then it is easy to conclude that
X=U[T−1X200]U∗, |
where X2∈Ct×(n−t).
Since
(Am)∗Am+1X=U[(Tm)∗0(Tm)∗(Nm)∗][Tm+1TTm+SNm0Nm+1][T−1X200]U∗=U[(Tm)∗Tm(Tm)∗Tm+1X2(Tm)∗Tm(Tm)∗Tm+1X2]U∗∈CHn, |
we obtain that X2=T−(m+1)Tm. Hence X=U[T−1T−(m+1)Tm00]U∗=AⓌm.
Motivated by the first two matrix equations XAk+1=Ak and XAX=X, we provide several characterizations of AⓌm.
Theorem 4.3. Let A∈Cn×nk, X∈Cn×n and let m∈Z+. Then the following conditions are equivalent:
(a) X=AⓌm.
(b) XAk+1=Ak, AX2=X, (Am)∗Am+1X∈CHn.
(c) XAk+1=Ak, AX2=X, Am+1X=PAkAm.
Proof. (a)⇔(b): This follows by Proposition 4.2 in [19].
(a)⇒(c): It is a direct consequence of Theorems 4.1 (c) and 4.2 (c).
(c)⇒(d): Assume that A is given by (2.1). By XAk+1=Ak, we get that
X=U[T−1X20X4]U∗, |
where X2∈Ct×(n−t) and X4∈C(n−t)×(n−t).
By AX2=X, we have that X4=NX42, which implies that
X4=NX42=N2X43=⋯=NkX4k+1=0. |
Using (2.12) and (2.13) and that Am+1X=PAkAm, we get
X=U[T−1T−(m+1)Tm00]U∗. |
Now, the proof follows directly.
(d)⇒(a): Since XAk+1=Ak, it follows that R(Ak)=R(XAk+1)⊆R(X) and by r(X)=r(Ak), we get R(Ak)=R(X). Hence, according to Theorem 4.1 (b), we get X=AⓌm.
According to Theorem 3.7, it follows that AX=PR(Ak),N((Ak)∗Am) and XA=PR(Ak),N((Ak)∗Am+1) when X=AⓌm. Conversely, the implication does not hold. Here's an example below.
Example 4.4. Let A=[I3L0N], X=[I3L0L], where N=[010001000], L=[001000000]. Then it is clear that k=Ind(A)=3 and AⓌ2=[I3L00]. It can be directly verified that AX=PR(A3),N((A3)∗A2),XA=PR(A3),N((A3)∗A3). However, X≠AⓌ2.
Based on the example above, the next theorem, we consider other characterizations of AⓌm by using AX=PR(Ak),N((Ak)∗Am) and XA=PR(Ak),N((Ak)∗Am+1).
Theorem 4.5. Let A∈Cn×nk be of the form (2.1), X∈Cn×n and m∈Z+. Then the following statements are equivalent:
(a) X=AⓌm;
(a) AX=PR(Ak),N((Ak)∗Am),XA=PR(Ak),N((Ak)∗Am+1) and r(X)=r(Ak);
(a) AX=PR(Ak),N((Ak)∗Am),XA=PR(Ak),N((Ak)∗Am+1) and XAX=X;
(a) AX=PR(Ak),N((Ak)∗Am),XA=PR(Ak),N((Ak)∗Am+1) and AX2=X.
Proof. (a)⇒(b): It is a direct consequence of Theorems 3.6 (a) and 3.7.
(b)⇒(c): Since XA=PR(Ak),N((Ak)∗Am+1) and r(X)=r(Ak), we get that R(X)=R(Ak) and by XA=PR(Ak),N((Ak)∗Am+1), we obtain XAX=X.
(c)⇒(d): From XA=PR(Ak),N((Ak)∗Am+1) and r(X)=r(Ak), we have that R(X)=R(Ak) and by AX=PR(Ak),N((Ak)∗Am), it follows that AX2=X.
(d)⇒(a): By XA=PR(Ak),N((Ak)∗Am+1) and AX2=X, it follows that
R(Ak)=R(XA)⊆R(X)=R(AX2)=⋯=R(AkXk+1)⊆R(Ak), |
which implies R(X)=R(Ak). By AX=PR(Ak),N((Ak)∗Am), we get that
(Ak)∗Am+1X=(Ak)∗Am. |
According to Theorem 4.1 (d), we have that X=AⓌm.
Analogously, we characterize AⓌm using that AX=PR(Ak),N((Ak)∗Am) or XA=PR(Ak),N((Ak)∗Am+1) as follows:
Theorem 4.6. Let A∈Cn×nk, X∈Cn×n and let m∈Z+. Then
(a) X=AⓌm is the unique solution of the system of equations:
AX=PR(Ak),N((Ak)∗Am), R(X)=R(Ak). | (4.1) |
(b) X=AⓌm is the unique solution of the system of equations:
XA=PR(Ak),N((Ak)∗Am+1), N(X)=N((Ak)∗Am). | (4.2) |
Proof. (a) By Theorems 3.6 (b) and 3.7 (a), it follows that X=AⓌm is a solution of the system of Eq (4.1). Conversely, if the system (4.1) is consistent, it follows that (Ak)∗AmAX=(Ak)∗Am. Hence by Theorem 4.1, X=AⓌm (d).
(b) By Theorems 3.6 (b) and 3.7 (b), it is evident that X=AⓌm is a solution of (4.2). Next, we prove the uniqueness of the solution.
Assume that X1,X2 satisfy the system of Eq (4.2). Then X1A=X2A and N(X1)=N(X2)=N((Ak)∗Am). Thus, we get that R(X∗1−X∗2)⊆N(A∗)⊆N((Ak)∗) and R(X∗1−X∗2)⊆R((Am)∗Ak). For any η∈N((Ak)∗)∩R((Am)∗Ak), we obtain that (Ak)∗η=0, η=(Am)∗Akξ for some ξ∈Cn. Since Ind(A)=k, we derive that R(Ak)=R(Ak+m), and it follows that Akξ=Ak+mξ0 for some ξ0∈Cn×n. Then we have that
0=(Ak)∗η=(Ak+m)∗Ak+mξ0. |
Premultiplying the equation above by ξ∗0, we derive that (Ak+mξ0)∗Ak+mξ0=0, which implies Ak+mξ0=0. Hence η=0, i.e., R(X∗1−X∗2)={0}, which implies X1=X2.
Remark 4.7. Notice that the condition R(X)=R(Ak) in Theorem 4.6 (a) can be replaced by R(X)⊆R(Ak). Also the condition N(X)=N((Ak)∗Am) in Theorem 4.6 (b) can be replaced by N(X)⊇N((Ak)∗Am).
From Theorem 3.1, we get an expression of AⓌm in terms of . In the next results, we present several expressions of AⓌm in terms of certain generalized inverses.
Theorem 5.1. Let A∈Cn×nk and let m∈Z+. Then the following statements hold:
(a) AⓌm=(AD)m+1PAkAm.
(c) AⓌm=(Ak)#Ak−m−1PAkAm (k≥m+1).
(d) AⓌm=(Am+1PAk)†Am.
(e) AⓌm=Am−1PAk(Am)Ⓦ.
Proof. Assume that A is given by (2.1). By (2.3)–(2.7) and (2.11)–(2.13), we get that
Am+1PAk=U[Tm+1000]U∗, | (5.1) |
(Am+1PAk)†=U[T−m−1000]U∗, | (5.2) |
(AD)m+1=U[T−m−1T−2−m−kTk00]U∗, | (5.3) |
(Ak)#=U[T−kT−2kTk00]U∗, | (5.4) |
![]() |
(5.5) |
(Am)Ⓦ=U[T−mT−2mTm00]U∗. | (5.6) |
(a) By (2.12), (2.13) and (5.3), it follows that
(AD)m+1PAkAm=U[T−1T−k−1Tk00]m+1[It000][TmTm0Nm]U∗=U[T−1T−(m+1)Tm00]U∗. |
Hence AⓌm=(AD)m+1PAkAm.
The proofs of (b)–(e) are analogous to that of (a).
Next, we consider the accuracy of the expression in Theorem 5.1 (a) for computing the m-weak group inverse.
Example 5.2. Let
![]() |
Assume that A is given by (2.1). Then
![]() |
It is clear that k = Ind(A) = 3. According to (2.12), (2.13), (3.2) and (5.3), a straightforward computation shows that
![]() |
Let K=(AD)3PA3A2. Then
![]() |
and
r1=∥AⓌ2−K∥=6.6885×10−14, |
where ∥⋅∥ is the Frobenius norm.
Hence, Theorem 5.1 (a) gives a good result in terms of computational accuracy.
In the following theorem, we present a connection between the (B,C)-inverse and the m-weak group inverse showing that the m-weak group inverse of A∈Cn×nk is its (Ak,(Ak)∗Am)-inverse.
Theorem 5.3. Let A∈Cn×nk and let m∈Z+. Then AⓌm=A(Ak,(Ak)∗Am).
Proof. By Theorem 3.7 we have that AⓌmAAk=Ak and ((Ak)∗Am)AAⓌm=(Ak)∗Am. From Theorem 3.6 (b), we derive that R(AⓌm)=R(Ak) and N(AⓌm)=N((Ak)∗Am). Evidently, AⓌm=A(Ak,(Ak)∗Am).
Now we will give some limit expressions of AⓌm, but before we need the next auxiliary lemma:
Lemma 5.4. [37] Let A∈Cm×n,X∈Cn×p and Y∈Cp×m. Then the following hold:
(a) limλ→0X(λIp+YAX)−1Y exists;
(b) r(XYAXY)=r(XY);
(c) A(2)R(XY),N(XY) exists,
in which case,
limλ→0X(λIp+YAX)−1Y=A(2)R(XY),N(XY). |
Theorem 5.5. Let A∈Cn×nk be given by (2.1) and let m∈N+. Then the following statements hold:
(a) AⓌm=limλ→0Ak(λIn+(Ak)∗Ak+m+1)−1(Ak)∗Am;
(b) AⓌm=limλ→0Ak(Ak)∗(λIn+Ak+m+1(Ak)∗)−1Am;
(c) AⓌm=limλ→0Ak(Ak)∗Am(λIn+Ak+1(Ak)∗Am)−1;
(d) AⓌm=limλ→0(λIn+Ak(Ak)∗Am+1)−1Ak(Ak)∗Am.
Proof. (a) It is easy to check that r(Ak(Ak)∗Am)=r((Ak)∗Am)=r(Ak)=t. By Theorem 3.6, we get that R(Ak)=R(Ak(Ak)∗Am), N((Ak)∗Am)=N(Ak(Ak)∗Am). From Theorem 3.6, we get
AⓌm=A(2)R(Ak),N((Ak)∗Am)=A(2)R(Ak(Ak)∗Am),N(Ak(Ak)∗Am). |
Let X=Ak,Y=(Ak)∗Am. By Lemma 5.4, we get that
AⓌm=limλ→0Ak(λIn+(Ak)∗Ak+m+1)−1(Ak)∗Am. |
The statements (b)–(d) can be similarly proved.
The following example will test the accuracy of expression in Theorem 5.5 (a) for computing the m-weak group inverse.
Example 5.6. Let
![]() |
with k = Ind(A) = 3. By , we get
![]() |
Together with (3.2), it follows that
![]() |
Let L=limλ→0A3(λIn+(A3)∗A6)−1(A3)∗A2. Then
![]() |
and
r2=∥AⓌ2−L∥=6.136×10−11, |
where ∥⋅∥ is the Frobenius norm. Hence, the representation in Theorem 5.5 (a) is efficient for computing the m-weak group inverse.
In this section, we consider some relations between the m-weak group inverse and other generalized inverses as well as certain matrix classes. The symbols COPn, CEPn, Ci−EPn and stand for the subsets of Cn×n consisting of orthogonal projectors (Hermitian idempotent matrices), EP (Range-Hermitian) matrices, i-EP matrices and k-core-EP matrices, respectively, i.e.,
![]() |
First, we will state the following lemma auxiliary lemma:
Lemma 6.1. Let A∈Cn×nk be given by (2.1). Then Tm=0 if and only if S=0.
Proof. Notice that Tm=0 can be equivalently expressed by the equation below:
Tm−1S+Tm−2SN+⋯+TSNm−2+SNm−1=0. | (6.1) |
Multiplying the equation above from the right side by Nk−1, we get SNk−1=0. Then multiplying from the right by Nk−2, we get SNk−1=0,. Similarly, we get SNk−3=0,⋯,SN=0. Now by (6.1), it follows that Tm−1S=0, i.e, S=0.
The next theorem provides some necessary and sufficient conditions for AⓌm to be equal to various transformations of A∈Cn×nk.
Theorem 6.2. Let A∈Cn×nk and let m∈Z+. Then the following statements hold:
(a) AⓌm∈A{1} if and only if A∈CCMn.
(b) AⓌm∈CCMn.
(c) AⓌm=A if and only if A=A3.
(d) AⓌm=A∗ if and only if AA∗∈COPn and A∈CEPn.
(e) AⓌm=PA if and only if A∈COPn.
Proof. Let A be given by (2.1).
(a) By (3.2), it follows that
AⓌm∈A{1}⟺AAⓌmA=A⟺U[TS+T−mTmN00]U∗=U[TS0N]U∗⟺N=0⟺A∈CCMn. |
(b) By (3.2), it is clear that r(AⓌm)=r((AⓌm)2)=t, which implies AⓌm∈CCMn.
(c) From (3.2), we get that
AⓌm=A⟺U[T−1(Tm+1)−1Tm00]U∗=U[TS0N]U∗⟺T2=It and N=0⟺A=A3. |
(d) According to (3.2), we obtain that
AⓌm=A∗⟺U[T−1(Tm+1)−1Tm00]U∗=U[T∗0S∗N∗]U∗⟺T−1=T∗,S=0 and N=0⟺AA∗∈COPnand A∈CEPn. |
(e) By (2.9) and (3.2), it follows that
AⓌm=PA⟺U[T−1(Tm+1)−1Tm00]U∗=U[It00NN†]U∗⟺T=It,NN†=0 and Tm=0⟺T=It,S=0 and N=0. |
Hence AⓌm=PA if and only if A∈COPn.
Using the core-EP decompositio, we proved that A∈Ci−EPn if and only if AⓌm∈CEPn. Therefore, we will consider certain equivalent conditions for AⓌm∈CEPn.
Lemma 6.3. [17] Let A∈Cn×nk be of the form (2.1). Then A∈Ci−EPn if and only if S=0.
Moreover, S=0 if and only if
Theorem 6.4. Let A∈Cn×nk and let m∈Z+. The following statements are equivalent:
(a) AⓌm∈CEPn;
(b) A∈Ci−EPn;
(c) AⓌ∈CEPn;
Proof. Let A∈Cn×nk be of the form (2.1). According to Lemma 6.3, we will prove that each of the statements (a), (c), (d) and (e) is equivalent to S=0.
(a) According to (3.2) and Lemma 6.1, it follows that
AⓌm∈CEPn⟺R(AⓌm)=R((AⓌm)∗)⟺(Tm+1)−1Tm=0⟺S=0. |
(c) By (2.4), we get that
AⓌ∈CEPn⟺R(AⓌ)=R((AⓌ)∗)⟺T−2S=0⟺S=0. |
(d) By (2.3), (3.2) and Lemma 6.1, it follows that
![]() |
(e) From (2.3), (3.2) and Lemma 6.1, we get that
![]() |
In [22], the authors proved that AⓌm=AD if and only if SNm=0. In the following results, we investigate the relation between the m-weak group inverse and other generalized inverses such as the MP-inverse, group inverse, core inverse, DMP-inverse, dual DMP-inverse, weak group inverse by core-EP decomposition.
Theorem 6.5. Let A∈Cn×nk be given by (2.1) and let m∈Z+. Then the following statements hold:
(a) AⓌm=A†⟺A∈CEPn;
(b) AⓌm=A#⟺A∈CCMn;
(d) AⓌm=AD,†⟺Tk−mTm=TkNN†;
(e) AⓌm=A†,D⟺SNm=0 and S=SN†N;
(f) AⓌm=AⓌ⟺SN=0 (m>1).
Proof. (a) It follows from (2.8) and (3.2) that
AⓌm=A†⟺U[T−1(Tm+1)−1Tm00]U∗=U[T∗△−T∗△SN†M∗△N†−M∗△SN†]U∗⟺M∗=0,N†=0,T−1=T∗△ and (Tm+1)−1Tm=−T∗△SN†⟺S=0 and N=0⟺A∈CEPn. |
(b) Since A# exits if and only if A∈CCMn, which is equivalent to N=0, we get by (2.6) and (3.2) the following:
AⓌm=A#⟺U[T−1(Tm+1)−1Tm00]U∗=U[T−1T−2S00]U∗ and N=0⟺(Tm+1)−1Tm=T−2S and N=0⟺N=0⟺A∈CCMn. |
(c) The proof follows similarly as in (b).
(d) Using (2.5) and (2.9) to AD,†=ADAA†, we derive
AD,†=[T−1(Tk+1)−1TkNN†00], |
and by (3.2), it follows that
AⓌm=AD,†⟺U[T−1(Tm+1)−1Tm00]U∗=U[T−1(Tk+1)−1TkNN†00]U∗⟺Tk−mTm=TkNN†. |
(e) Using (2.5) and (2.10) and the faact that A†,D=A†AAD, we obtain that
A†,D=[T∗△−T∗△T−kTkM∗△M∗△T−kTk], |
which together with (3.2), gives
AⓌm=A†,D⟺U[T−1(Tm+1)−1Tm00]U∗=U[T∗△T∗△T−kTkM∗△M∗△T−kTk]U∗⟺M∗=0,T−1=T∗△ and (Tm+1)−1Tm=T∗△T−kTk⟺S=SN†N and Tk−mTm=Tk⟺S=SN†N and SNm=0. |
(f) If m>1, from (2.4) and (3.2), we get
AⓌm=AⓌ⟺U[T−1(Tm+1)−1Tm00]U∗=U[T−1T−2S00]U∗⟺(Tm+1)−1Tm=T−2S. |
Clearly, (Tm+1)−1Tm=T−2S is equivalent to T−3SN+⋯+(Tm+1)−1SNm−1=0, which is further equivalent to SN=0. Hence AⓌm=AⓌ if and only if SN=0.
In this section, we consider a relation between the m-weak group inverse and the nonsingular bordered matrix, which will be applied to the Cramer's rule for the solution of the restricted matrix equation.
Theorem 7.1. Let A∈Cn×nk be such that r(Ak)=t and let m∈Z+. Let B∈Cn×(n−t) and C∗∈Cn×(n−t) be of full column rank such that N((Ak)∗Am)=R(B) and R(Ak)=N(C). Then the bordered matrix
K=[ABC0] |
is invertible and its inverse is given by
K−1=[AⓌm (In−AⓌmA)C†B†(In−AAⓌm) B†(AAⓌmA−A)C†]. |
Proof. Let X=[AⓌm (In−AⓌmA)C†B†(In−AAⓌm) B†(AAⓌmA−A)C†]. Since R(AⓌm)=R(Ak)=N(C), we have that CAⓌm=0. Since C is a full row rank matrix, it is right invertible and CC†=In−t. From
R(In−AAⓌm)=N(AAⓌm)=N((Ak)∗Am)=R(B)=R(BB†), |
we get BB†(In−AAⓌm)=In−AAⓌm. Hence,
KX=[AAⓌm+BB†(In−AAⓌm) A(In−AⓌmA)C†+BB†(AAⓌmA−A)C†CAⓌm C(In−AⓌmA)C†]=[AAⓌm+In−AAⓌm A(In−AⓌmA)C†−(In−AAⓌm)AC†0 CC†]=[In 00 In−t]. |
Thus, X=K−1.
In the next result, we will discuss the solution of the restricted matrix equation
AX=D, R(X)⊆R(Ak), | (7.1) |
using the m-weak group inverse.
Theorem 7.2. Let A∈Cn×nk, X∈Cn×p and D∈Cn×p. If R(D)⊆R(Ak), then the restricted matrix equation
AX=D, R(X)⊆R(Ak) | (7.2) |
has a unique solution X=AⓌmD.
Proof. Since R(Ak)=R(AAk) and R(D)⊆R(Ak), we get that R(D)⊆AR(Ak), which implies solvability of the matrix Eq (7.1). Obviously, X=AⓌmD is a solution of (7.1). Then we prove the uniqueness of X. If X1 also satisfies (7.1), then
X=AⓌmD=AⓌmAX1=PR(Ak),N((Ak)∗Am)X1=X1. |
Based on the nonsingularity of the bordered matrix given in Theorem 7.1, we will show in the next theorem how the Cramer's rule can be used for solving the restricted matrix Eq (7.1).
Theorem 7.3. Let A∈Cn×nk be such that r(Ak)=t and let X∈Cn×p and D∈Cn×p. Let B∈Cn×(n−t) and C∗∈Cn×(n−t) be full column rank matrices such that N((Ak)∗Am)=R(B) and R(Ak)=N(C). Then the unique solution of the restricted matrix Eq (7.1) is given by X=[xij], where
xij=det[A(i→dj)BC(i→0)0]det[ABC0], i=1,2,...,n,j=1,2,...,m, | (7.3) |
where dj denotes the j-th column of D.
Proof. Since X is the solution of the restricted matrix Eq (7.1), we get that R(X)⊆R(Ak)=N(C), which implies CX=0. Then the restricted matrix Eq (7.1) can be rewritten as
[ABC0][X0]=[AXCX]=[D0]. |
By Theorem 7.1, we have that [ABC0] is invertible. Consequently, (7.2) follows from the Cramer's rule for the above equation.
Example 7.4. Let
A=[100100010010001001000010000001000000], D=[1014242861920224101415000000000000],B=[123012136−2−4−7124−1−3−6], C=[000106000120000003]. |
It can be verified that Ind(A)=3. Then we get that
![]() |
It is easy to check that
X=AⓌ2D=[1014242861920224101415000000000000] |
satisfies the restricted matrix equation AX=D and R(X)⊆R(A3). By simple calculations, we can also get that the components of X can be expressed by (7.2).
This paper gives a new definition of the m-weak group inverse for the complex matrices, which extends the Drazin inverse and the weak group inverse. Some characterizations of the m-weak group inverse in terms of the range space, null space, rank, and projectors are presented. Several representations of the m-weak group inverse involving some known generalized inverses as well as limitations are also derived. The representation in Theorem 5.1 gives a better result in terms of the computational accuracy (see Examples 5.2 and 5.6). The m-weak group inverses are concerned with the solution of a restricted matrix Eq (7.1). The solution of (7.1) can also be expressed by the Cramer's rule (see Theorem 7.3). In [38,39,40], there are some iterative methods and algorithms to compute the outer inverses. Motivated by these, further investigations deserve more attention as follows:
(1) The applications of the m-weak group inverse in linear equations and matrix equations;
(2) Perturbation formulae as well as perturbation bounds for the m-weak group inverse;
(3) Iterative algorithm, a splitting method for computing the m-weak group inverse;
(4) Other representations of the m-weak group inverse.
This research is supported by the National Natural Science Foundation of China (No. 11961076).
The authors declare no conflict of interest.
[1] |
Puetz J, Wurm FM (2019) Recombinant proteins for industrial versus pharmaceutical purposes: a review of process and pricing. Processes 7: 476. doi: 10.3390/pr7080476
![]() |
[2] |
Zhang W, Yang Y, Liu X, et al. (2019) Development of a secretory expression system with high compatibility between expression elements and an optimized host for endoxylanase production in Corynebacterium glutamicum. Microb Cell Fact 18: 72. doi: 10.1186/s12934-019-1116-y
![]() |
[3] |
Sadeghian-Rizi T, Ebrahimi A, Moazzen F, et al. (2019) Improvement of solubility and yield of recombinant protein expression in E. coli using a two-step system. Res Pharm Sci 14: 400-407. doi: 10.4103/1735-5362.268200
![]() |
[4] |
Mikiewicz D, Plucienniczak A, Krzysik-Bierczynska A, et al. (2019) Novel expression vectors based on the pIGDM1 plasmid. Mol Biotechnol 61: 763-773. doi: 10.1007/s12033-019-00201-6
![]() |
[5] |
Duggar BM (1948) Aureomycin: a product of the continuing search for new antibiotics. Ann NY Acad Sci 51: 177-181. doi: 10.1111/j.1749-6632.1948.tb27262.x
![]() |
[6] |
Nagarajan R, Boeck LD, Gorman M, et al. (1971) Beta-lactam antibiotics from Streptomyces. J Am Chem Soc 93: 2308-2310. doi: 10.1021/ja00738a035
![]() |
[7] |
Waksman SA, Lechevalier HA (1949) Neomycin, a new antibiotic active against streptomycin-resistant bacteria, including tuberculosis organisms. Science 109: 305-307. doi: 10.1126/science.109.2830.305
![]() |
[8] |
Waksman SA (1953) Streptomycin: background, isolation, properties, and utilization. Science 118: 259-266. doi: 10.1126/science.118.3062.259
![]() |
[9] |
Vining LC, Shapiro S (1984) Chloramphenicol production in carbon-limited media: effect of methyl α-glucoside. J Antibiot 37: 74-76. doi: 10.7164/antibiotics.37.74
![]() |
[10] |
De Lima Procópio RE, Da Silva IR, Martins MK, et al. (2012) Antibiotics produced by Streptomyces. Braz J Infect Dis 16: 466-471. doi: 10.1016/j.bjid.2012.08.014
![]() |
[11] |
Baltz RH (2010) Streptomyces and Saccharopolyspora hosts for heterologous expression of secondary metabolite gene clusters. J Ind Microbiol Biotechnol 37: 759-772. doi: 10.1007/s10295-010-0730-9
![]() |
[12] |
Raja A, Prabakarana P (2011) Actinomycetes and drug-an overview. Am J Drug Discov Dev 1: 75-84. doi: 10.3923/ajdd.2011.75.84
![]() |
[13] |
Gamboa-Suasnavart RA, Marín-Palacio LD, Martínez-Sotelo JA, et al. (2013) Scale-up from shake flasks to bioreactor, based on power input and Streptomyces lividans morphology, for the production of recombinant APA (45/47 kDa protein) from Mycobacterium tuberculosis. World J Microbiol Biotechnol 29: 1421-1429. doi: 10.1007/s11274-013-1305-5
![]() |
[14] |
Muhamadali H, Xu Y, Ellis DI, et al. (2015) Metabolomics investigation of recombinant mTNFα production in Streptomyces lividans. Microb Cell Fact 14: 157-157. doi: 10.1186/s12934-015-0350-1
![]() |
[15] |
Hamed MB, Karamanou S, Olafsdottir S, et al. (2017) Large-scale production of a thermostable Rhodothermus marinus cellulase by heterologous secretion from Streptomyces lividans. Microb Cell Fact 16: 232. doi: 10.1186/s12934-017-0847-x
![]() |
[16] |
Anné J, Maldonado B, Van Impe J, et al. (2012) Recombinant protein production and Streptomycetes. J Biotechnol 158: 159-167. doi: 10.1016/j.jbiotec.2011.06.028
![]() |
[17] |
Daniels W, Bouvin J, Busche T, et al. (2018) Transcriptomic and fluxomic changes in Streptomyces lividans producing heterologous protein. Microb Cell Fact 17: 198. doi: 10.1186/s12934-018-1040-6
![]() |
[18] | Simonen M, Palva I (1993) Protein secretion in Bacillus species. Microbiol Mol Biol Rev 57: 109-137. |
[19] |
Wu SC, Ye R, Wu XC, et al. (1998) Enhanced secretory production of a single-chain antibody fragment from Bacillus subtilis by coproduction of molecular chaperones. J Bacteriol 180: 2830-2835. doi: 10.1128/JB.180.11.2830-2835.1998
![]() |
[20] |
Wu SC, Yeung JC, Duan Y, et al. (2002) Functional production and characterization of a fibrin-specific single-chain antibody fragment from Bacillus subtilis: effects of molecular chaperones and a wall-bound protease on antibody fragment production. Appl Environ Microbiol 68: 3261-3269. doi: 10.1128/AEM.68.7.3261-3269.2002
![]() |
[21] |
Lakowitz A, Krull R, Biedendieck R (2017) Recombinant production of the antibody fragment D1.3 scFv with different Bacillus strains. Microb Cell Fact 16: 14. doi: 10.1186/s12934-017-0625-9
![]() |
[22] |
Inoue Y, Ohta T, Tada H, et al. (1997) Efficient production of a functional mouse/human chimeric Fab′ against human urokinase-type plasminogen activator by Bacillus brevis. Appl Microbiol Biotechnol 48: 487-492. doi: 10.1007/s002530051084
![]() |
[23] |
Shiroza T, Shinozaki-Kuwahara N, Hayakawa M, et al. (2003) Production of a single-chain variable fraction capable of inhibiting the Streptococcus mutans glucosyltransferase in Bacillus brevis: construction of a chimeric shuttle plasmid secreting its gene product. Biochim Biophys Acta 1626: 57-64. doi: 10.1016/S0167-4781(03)00038-1
![]() |
[24] |
Jordan E, Hust M, Roth A, et al. (2007) Production of recombinant antibody fragments in Bacillus megaterium. Microb Cell Fact 6: 2. doi: 10.1186/1475-2859-6-2
![]() |
[25] |
Wang H, Zhang X, Qiu J, et al. (2019) Development of Bacillus amyloliquefaciens as a high-level recombinant protein expression system. J Ind Microbiol Biotechnol 46: 113-123. doi: 10.1007/s10295-018-2089-2
![]() |
[26] |
Zhang XZ, Cui ZL, Hong Q, et al. (2005) High-level expression and secretion of methyl parathion hydrolase in Bacillus subtilis WB800. Appl Environ Microbiol 71: 4101-4103. doi: 10.1128/AEM.71.7.4101-4103.2005
![]() |
[27] |
Contesini FJ, De Melo RR, Sato HH (2018) An overview of Bacillus proteases: from production to application. Crit Rev Biotechnol 38: 321-334. doi: 10.1080/07388551.2017.1354354
![]() |
[28] |
Cai D, Rao Y, Zhan Y, et al. (2019) Engineering Bacillus for efficient production of heterologous protein: current progress, challenge and prospect. J Appl Microbiol 126: 1632-1642. doi: 10.1111/jam.14192
![]() |
[29] |
Song AAL, In LLA, Lim SHE, et al. (2017) A review on Lactococcus lactis: from food to factory. Microb Cell Fact 16: 55. doi: 10.1186/s12934-017-0669-x
![]() |
[30] |
Boumaiza M, Colarusso A, Parrilli E, et al. (2018) Getting value from the waste: recombinant production of a sweet protein by Lactococcus lactis grown on cheese whey. Microb Cell Fact 17: 126. doi: 10.1186/s12934-018-0974-z
![]() |
[31] |
Singh SK, Tiendrebeogo RW, Chourasia BK, et al. (2018) Lactococcus lactis provides an efficient platform for production of disulfide-rich recombinant proteins from Plasmodium falciparum. Microb Cell Fact 17: 55. doi: 10.1186/s12934-018-0902-2
![]() |
[32] |
Rezaei M, Rabbani Khorasgani M, Zarkesh Esfahani SH, et al. (2019) Production of Brucella melitensis Omp16 protein fused to the human interleukin 2 in Lactococcus lactis MG1363 toward developing a Lactococcus-based vaccine against brucellosis. Can J Microbiol 66: 39-45. doi: 10.1139/cjm-2019-0261
![]() |
[33] |
Lahiri A, Sharif S, Mallick AI (2019) Intragastric delivery of recombinant Lactococcus lactis displaying ectodomain of influenza matrix protein 2 (M2e) and neuraminidase (NA) induced focused mucosal and systemic immune responses in chickens. Mol Immunol 114: 497-512. doi: 10.1016/j.molimm.2019.08.015
![]() |
[34] |
Song S, Li P, Zhang R, et al. (2019) Oral vaccine of recombinant Lactococcus lactis expressing the VP1 protein of duck hepatitis A virus type 3 induces mucosal and systemic immune responses. Vaccine 37: 4364-4369. doi: 10.1016/j.vaccine.2019.06.026
![]() |
[35] | Ghasemi Z, Varasteh AR, Moghadam M, et al. (2018) Production of recombinant protein of Salsola kali (Sal k1) pollen allergen in Lactococcus lactis. Iran J Allergy Asthma Immunol 17: 134-143. |
[36] |
Taghinezhad-S S, Mohseni AH, Keyvani H, et al. (2019) Protection against human papillomavirus type 16-induced tumors in C57BL/6 mice by mucosal vaccination with Lactococcus lactis NZ9000 expressing E6 oncoprotein. Microb Pathog 126: 149-156. doi: 10.1016/j.micpath.2018.10.043
![]() |
[37] |
Loh JMS, Lorenz N, Tsai CJY, et al. (2017) Mucosal vaccination with pili from Group A Streptococcus expressed on Lactococcus lactis generates protective immune responses. Sci Rep 7: 7174. doi: 10.1038/s41598-017-07602-0
![]() |
[38] |
Joseph BC, Pichaimuthu S, Srimeenakshi S, et al. (2015) An overview of the parameters for recombinant protein expression in Escherichia coli. J Cell Sci Ther 6: 1000221. doi: 10.4172/2157-7013.1000221
![]() |
[39] |
Itakura K, Hirose T, Crea R, et al. (1977) Expression in Escherichia coli of a chemically synthesized gene for the hormone somatostatin. Science 198: 1056-1063. doi: 10.1126/science.412251
![]() |
[40] |
Hayat SMG, Farahani N, Golichenati B, et al. (2018) Recombinant protein expression in Escherichia coli (E.coli): what we need to know. Curr Pharm Des 24: 718-725. doi: 10.2174/1381612824666180131121940
![]() |
[41] | Tripathi NK (2016) Production and purification of recombinant proteins from Escherichia coli. Chem Bio Eng Rev 3: 116-133. |
[42] |
Liu HL, Yang SJ, Liu Q, et al. (2018) A process for production of trehalose by recombinant trehalose synthase and its purification. Enzyme Microb Technol 113: 83-90. doi: 10.1016/j.enzmictec.2017.11.008
![]() |
[43] |
Celesia D, Salzmann I, Porto EV, et al. (2017) Production of a recombinant catechol 2,3-dioxygenase for the degradation of micropollutants. CHIMIA Int J Chem 71: 734-738. doi: 10.2533/chimia.2017.734
![]() |
[44] |
Tajbakhsh M, Akhavan MM, Fallah F, et al. (2018) A recombinant snake cathelicidin derivative peptide: antibiofilm properties and expression in Escherichia coli. Biomolecules 8: 118. doi: 10.3390/biom8040118
![]() |
[45] |
Jin H, Cantin GT, Maki S, et al. (2011) Soluble periplasmic production of human granulocyte colony-stimulating factor (G-CSF) in Pseudomonas fluorescens. Protein Expres Purif 78: 69-77. doi: 10.1016/j.pep.2011.03.002
![]() |
[46] |
Retallack DM, Schneider JC, Mitchell J, et al. (2007) Transport of heterologous proteins to the periplasmic space of Pseudomonas fluorescens using a variety of native signal sequences. Biotechnol Lett 29: 1483-1491. doi: 10.1007/s10529-007-9415-5
![]() |
[47] |
Chen R (2012) Bacterial expression systems for recombinant protein production: E. coli and beyond. Biotechnol Adv 30: 1102-1107. doi: 10.1016/j.biotechadv.2011.09.013
![]() |
[48] |
Hermann T (2003) Industrial production of amino acids by coryneform bacteria. J Biotechnol 104: 155-172. doi: 10.1016/S0168-1656(03)00149-4
![]() |
[49] |
Kondoh M, Hirasawa T (2019) L-cysteine production by metabolically engineered Corynebacterium glutamicum. Appl Microbiol Biotechnol 103: 2609-2619. doi: 10.1007/s00253-019-09663-9
![]() |
[50] |
Liu X, Zhang W, Zhao Z, et al. (2017) Protein secretion in Corynebacterium glutamicum. Crit Rev Biotechnol 37: 541-551. doi: 10.1080/07388551.2016.1206059
![]() |
[51] |
Date M, Itaya H, Matsui H, et al. (2006) Secretion of human epidermal growth factor by Corynebacterium glutamicum. Lett Appl Microbiol 42: 66-70. doi: 10.1111/j.1472-765X.2005.01802.x
![]() |
[52] |
Lee MJ, Kim P (2018) Recombinant protein expression system in Corynebacterium glutamicum and its application. Front Microbiol 9: 2523. doi: 10.3389/fmicb.2018.02523
![]() |
[53] |
Haas T, Graf M, Nieß A, et al. (2019) Identifying the growth modulon of Corynebacterium glutamicum. Front Microbiol 10: 974. doi: 10.3389/fmicb.2019.00974
![]() |
[54] |
Felpeto-Santero C, Galan B, Luengo JM, et al. (2019) Identification and expression of the 11β-steroid hydroxylase from Cochliobolus lunatus in Corynebacterium glutamicum. Microb Biotechnol 12: 856-868. doi: 10.1111/1751-7915.13428
![]() |
[55] |
Overton TW (2014) Recombinant protein production in bacterial hosts. Drug Discov Today 19: 590-601. doi: 10.1016/j.drudis.2013.11.008
![]() |
[56] |
Jin LQ, Jin WR, Ma ZC, et al. (2019) Promoter engineering strategies for the overproduction of valuable metabolites in microbes. Appl Microbiol Biotechnol 103: 8725-8736. doi: 10.1007/s00253-019-10172-y
![]() |
[57] |
Goldstein MA, Doi RH (1995) Prokaryotic promoters in biotechnology. Biotechnology Annual Review Amsterdam: Elsevier, 105-128. doi: 10.1016/S1387-2656(08)70049-8
![]() |
[58] |
Chaudhary AK, Lee EY (2015) Tightly regulated and high level expression vector construction for Escherichia coli BL21 (DE3). J Ind Eng Chem 31: 367-373. doi: 10.1016/j.jiec.2015.07.011
![]() |
[59] |
Trung NT, Hung NM, Thuan NH, et al. (2019) An auto-inducible phosphate-controlled expression system of Bacillus licheniformis. BMC Biotechnol 19: 3. doi: 10.1186/s12896-018-0490-6
![]() |
[60] |
Meyers A, Furtmann C, Gesing K, et al. (2019) Cell density—dependent auto—inducible promoters for expression of recombinant proteins in Pseudomonas putida. Microb Biotechnol 12: 1003-1013. doi: 10.1111/1751-7915.13455
![]() |
[61] |
Brunner M, Bujard H (1987) Promoter recognition and promoter strength in the Escherichia coli system. EMBO J 6: 3139-3144. doi: 10.1002/j.1460-2075.1987.tb02624.x
![]() |
[62] |
Shimada T, Yamazaki Y, Tanaka K, et al. (2014) The whole set of constitutive promoters recognized by RNA polymerase RpoD holoenzyme of Escherichia coli. PLoS One 9: e90447. doi: 10.1371/journal.pone.0090447
![]() |
[63] |
Pavco PA, Steege DA (1991) Characterization of elongating T7 and SP6 RNA polymerases and their response to a roadblock generated by a site-specific DNA binding protein. Nucleic Acids Res 19: 4639-4646. doi: 10.1093/nar/19.17.4639
![]() |
[64] |
Vavrová Ľ, Muchová K, Barák I (2010) Comparison of different Bacillus subtilis expression systems. Res Microbiol 161: 791-797. doi: 10.1016/j.resmic.2010.09.004
![]() |
[65] |
Retnoningrum DS, Santika IWM, Kesuma S, et al. (2019) Construction and characterization of a medium copy number expression vector carrying auto-inducible dps promoter to overproduce a bacterial superoxide dismutase in Escherichia coli. Mol Biotechnol 61: 231-240. doi: 10.1007/s12033-018-00151-5
![]() |
[66] |
Jaishankar J, Srivastava P (2017) Molecular basis of stationary phase survival and applications. Front Microbiol 8: 2000. doi: 10.3389/fmicb.2017.02000
![]() |
[67] |
Xu J, Liu X, Yu X, et al. (2020) Identification and characterization of sequence signatures in the Bacillus subtilis promoter Pylb for tuning promoter strength. Biotechnol Lett 42: 115-124. doi: 10.1007/s10529-019-02749-4
![]() |
[68] |
Zhou C, Ye B, Cheng S, et al. (2019) Promoter engineering enables overproduction of foreign proteins from a single copy expression cassette in Bacillus subtilis. Microb Cell Fact 18: 111. doi: 10.1186/s12934-019-1159-0
![]() |
[69] |
Yu X, Xu J, Liu X, et al. (2015) Identification of a highly efficient stationary phase promoter in Bacillus subtilis. Sci Rep 5: 18405. doi: 10.1038/srep18405
![]() |
[70] |
Ma Y, Cui Y, Du L, et al. (2018) Identification and application of a growth-regulated promoter for improving L-valine production in Corynebacterium glutamicum. Microb Cell Fact 17: 185. doi: 10.1186/s12934-018-1031-7
![]() |
[71] |
Brewster RC, Jones DL, Phillips R (2012) Tuning promoter strength through RNA polymerase binding site design in Escherichia coli. PLoS Comput Biol 8: e1002811. doi: 10.1371/journal.pcbi.1002811
![]() |
[72] |
Hannig G, Makrides SC (1998) Strategies for optimizing heterologous protein expression in Escherichia coli. Trends Biotechnol 16: 54-60. doi: 10.1016/S0167-7799(97)01155-4
![]() |
[73] |
Tegel H, Ottosson J, Hober S (2011) Enhancing the protein production levels in Escherichia coli with a strong promoter. FEBS J 278: 729-739. doi: 10.1111/j.1742-4658.2010.07991.x
![]() |
[74] |
Seyfi R, Babaeipour V, Mofid MR, et al. (2019) Expression and production of recombinant scorpine as a potassium channel blocker protein in Escherichia coli. Biotechnol Appl Bioc 66: 119-129. doi: 10.1002/bab.1704
![]() |
[75] |
Qaiser H, Aslam F, Iftikhar S, et al. (2018) Construction and recombinant expression of Pseudomonas aeruginosa truncated exotoxin A in Escherichia coli. Cell Mol Biol 64: 64-69. doi: 10.14715/cmb/2018.64.1.12
![]() |
[76] |
Khushoo A, Pal Y, Mukherjee KJ (2005) Optimization of extracellular production of recombinant asparaginase in Escherichia coli in shake-flask and bioreactor. Appl Microbiol Biotechnol 68: 189-197. doi: 10.1007/s00253-004-1867-0
![]() |
[77] |
Przystałowska H, Zeyland J, Kosmider A, et al. (2015) 1, 3-propanediol production by Escherichia coli using genes from Citrobacter freundii atcc 8090. Acta Biochim Pol 62: 589-597. doi: 10.18388/abp.2015_1061
![]() |
[78] |
Liang Q, Zhang H, Li S, et al. (2011) Construction of stress-induced metabolic pathway from glucose to 1,3-propanediol in Escherichia coli. Appl Microbiol Biotechnol 89: 57-62. doi: 10.1007/s00253-010-2853-3
![]() |
[79] |
Hjelm A, Karyolaimos A, Zhang Z, et al. (2017) Tailoring Escherichia coli for the L-rhamnose Pbad promoter-based production of membrane and secretory proteins. ACS Synth Biol 6: 985-994. doi: 10.1021/acssynbio.6b00321
![]() |
[80] |
Nguyen NH, Kim JR, Park S (2019) Development of biosensor for 3-hydroxypropionic acid. Biotechnol Bioproc E 24: 109-118. doi: 10.1007/s12257-018-0380-8
![]() |
[81] |
Liu Q, Ouyanga S, Kim J, et al. (2007) The impact of PHB accumulation on L-glutamate production by recombinant Corynebacterium glutamicum. J Biotechnol 132: 273-279. doi: 10.1016/j.jbiotec.2007.03.014
![]() |
[82] |
Skerra A (1994) Use of the tetracycline promoter for the tightly regulated production of a murine antibody fragment in Escherichia coli. Gene 151: 131-135. doi: 10.1016/0378-1119(94)90643-2
![]() |
[83] |
Suzuki N, Watanabe K, Okibe N, et al. (2008) Identification of new secreted proteins and secretion of heterologous amylase by C. glutamicum. Appl Microbiol Biotechnol 82: 491-500. doi: 10.1007/s00253-008-1786-6
![]() |
[84] |
Donovan RS, Robinson CW, Glick BR (2000) Optimizing the expression of a monoclonal antibody fragment under the transcriptional control of the Escherichia coli lac promoter. Can J Microbiol 46: 532-541. doi: 10.1139/w00-026
![]() |
[85] | Jia H, Li H, Zhang L, et al. (2018) Development of a novel gene expression system for secretory production of heterologous proteins via the general secretory (sec) pathway in Corynebacterium glutamicum. Iran J Biotechnol 16: e1746. |
[86] |
Caspeta L, Flores N, Pérez NO, et al. (2009) The effect of heating rate on Escherichia coli metabolism, physiological stress, transcriptional response, and production of temperature-induced recombinant protein: a scale-down study. Biotechnol Bioeng 102: 468-482. doi: 10.1002/bit.22084
![]() |
[87] |
Wild J, Hradecna Z, Szybalski W (2002) Conditionally amplifiable BACs: switching from single-copy to high-copy vectors and genomic clones. Genome Res 12: 1434-1444. doi: 10.1101/gr.130502
![]() |
[88] |
Paek A, Kim MJ, Park HY, et al. (2020) Functional expression of recombinant hybrid enzymes composed of bacterial and insect's chitinase domains in E. coli. Enzyme Microb Technol 136: 109492. doi: 10.1016/j.enzmictec.2019.109492
![]() |
[89] |
Liu H, Wang S, Song L, et al. (2019) Trehalose production using recombinant trehalose synthase in Bacillus subtilis by integrating fermentation and biocatalysis. J Agric Food Chem 67: 9314-9324. doi: 10.1021/acs.jafc.9b03402
![]() |
[90] |
Phan HTT, Nhi NNY, Tien LT, et al. (2019) Construction of expression plasmid for Bacillus subtilis using Pspac promoter and BgaB as a reporter. Sci Technol Dev J 22: 239-246. doi: 10.32508/stdj.v22i2.1284
![]() |
[91] |
Jørgensen CM, Vrang A, Madsen SM (2014) Recombinant protein expression in Lactococcus lactis using the P170 expression system. FEMS Microbiol Lett 351: 170-178. doi: 10.1111/1574-6968.12351
![]() |
[92] |
Hanif MU, Gul R, Hanif MI, et al. (2017) Heterologous secretory expression and characterization of dimerized bone morphogenetic protein 2 in Bacillus subtilis. Biomed Res Int 2017: 9350537. doi: 10.1155/2017/9350537
![]() |
[93] |
Hallewell RA, Emtage S (1980) Plasmid vectors containing the tryptophan operon promoter suitable for efficient regulated expression of foreign genes. Gene 9: 27-47. doi: 10.1016/0378-1119(80)90165-1
![]() |
[94] |
Han L, Cui W, Suo F, et al. (2019) Development of a novel strategy for robust synthetic bacterial promoters based on a stepwise evolution targeting the spacer region of the core promoter in Bacillus subtilis. Microb Cell Fact 18: 96. doi: 10.1186/s12934-019-1148-3
![]() |
[95] |
Pothoulakis G, Ellis T (2018) Construction of hybrid regulated mother-specific yeast promoters for inducible differential gene expression. PLoS One 13: e0194588. doi: 10.1371/journal.pone.0194588
![]() |
[96] |
Zhang M, Li F, Marquez-Lago TT, et al. (2019) MULTiPly: a novel multi-layer predictor for discovering general and specific types of promoters. Bioinformatics 35: 2957-2965. doi: 10.1093/bioinformatics/btz016
![]() |
[97] |
Liu X, Yang H, Zheng J, et al. (2017) Identification of strong promoters based on the transcriptome of Bacillus licheniformis. Biotechnol Lett 39: 873-881. doi: 10.1007/s10529-017-2304-7
![]() |
[98] |
Yuan F, Li K, Zhou C, et al. (2020) Identification of two novel highly inducible promoters from Bacillus licheniformis by screening transcriptomic data. Genomics 112: 1866-1871. doi: 10.1016/j.ygeno.2019.10.021
![]() |
[99] | Wang Y, Wang H, Liu L, et al. (2019) Synthetic promoter design in Escherichia coli based on generative adversarial network. BioRxiv 2019: 563775. |
[100] |
Presnell KV, Flexer-Harrison M, Alper HS (2019) Design and synthesis of synthetic UP elements for modulation of gene expression in Escherichia coli. Synth Syst Biotechnol 4: 99-106. doi: 10.1016/j.synbio.2019.04.002
![]() |
[101] |
Rhodius VA, Mutalik VK (2010) Predicting strength and function for promoters of the Escherichia coli alternative sigma factor, σE. PNAS 107: 2854-2859. doi: 10.1073/pnas.0915066107
![]() |
[102] |
Meng F, Zhu X, Nie T, et al. (2018) Enhanced expression of pullulanase in Bacillus subtilis by new strong promoters mined from transcriptome data, both alone and in combination. Front Microbiol 9: 2635. doi: 10.3389/fmicb.2018.02635
![]() |
[103] |
Hou Y, Chen S, Wang J, et al. (2019) Isolating promoters from Corynebacterium ammoniagenes ATCC 6871 and application in CoA synthesis. BMC Biotechnol 19: 76. doi: 10.1186/s12896-019-0568-9
![]() |
[104] |
Wang J, Ai X, Mei H, et al. (2013) High-throughput identification of promoters and screening of highly active promoter-5′-UTR DNA region with different characteristics from Bacillus thuringiensis. PLoS One 8: e62960. doi: 10.1371/journal.pone.0062960
![]() |
[105] |
Yim SS, An SJ, Kang M, et al. (2013) Isolation of fully synthetic promoters for high-level gene expression in Corynebacterium glutamicum. Biotechnol Bioeng 110: 2959-2969. doi: 10.1002/bit.24954
![]() |
[106] |
Nie Z, Luo H, Li J, et al. (2020) High-throughput screening of T7 promoter mutants for soluble expression of cephalosporin C acylase in E. coli. Appl Biochem Biotechnol 190: 293-304. doi: 10.1007/s12010-019-03113-y
![]() |
[107] |
Yang S, Liu Q, Zhang Y, et al. (2018) Construction and characterization of broad-spectrum promoters for synthetic biology. ACS Synth Biol 7: 287-291. doi: 10.1021/acssynbio.7b00258
![]() |
[108] |
Yang J, Ruff AJ, Hamer SN, et al. (2016) Screening through the PLICable promoter toolbox enhances protein production in Escherichia coli. Biotechnol J 11: 1639-1647. doi: 10.1002/biot.201600270
![]() |
[109] |
Aoki S, Kondo T, Ishiura M (2002) A promoter-trap vector for clock-controlled genes in the cyanobacterium Synechocystis sp. PCC 6803. J Microbiol Methods 49: 265-274. doi: 10.1016/S0167-7012(01)00376-1
![]() |
[110] |
Yang M, Zhang W, Ji S, et al. (2013) Generation of an artificial double promoter for protein expression in Bacillus subtilis through a promoter trap system. PLoS one 8: e56321. doi: 10.1371/journal.pone.0056321
![]() |
[111] |
Yim SS, An SJ, Choi JW, et al. (2014) High-level secretory production of recombinant single-chain variable fragment (scFv) in Corynebacterium glutamicum. Appl Microbiol Biotechnol 98: 273-284. doi: 10.1007/s00253-013-5315-x
![]() |
[112] |
Morowvat MH, Babaeipour V, Memari HR, et al. (2015) Optimization of fermentation conditions for recombinant human interferon beta production by Escherichia coli using the response surface methodology. Jundishapur J Microbiol 8: e16236. doi: 10.5812/jjm.8(4)2015.16236
![]() |
[113] |
Sevillano L, Vijgenboom E, van Wezel GP, et al. (2016) New approaches to achieve high level enzyme production in Streptomyces lividans. Microb Cell Fact 15: 28. doi: 10.1186/s12934-016-0425-7
![]() |
[114] |
Shen R, Yin J, Ye JW, et al. (2018) Promoter engineering for enhanced P (3HB-co-4HB) production by Halomonas bluephagenesis. ACS Synth Biol 7: 1897-1906. doi: 10.1021/acssynbio.8b00102
![]() |
[115] |
Gawin A, Peebo K, Hans S, et al. (2019) Construction and characterization of broad-host-range reporter plasmid suitable for on-line analysis of bacterial host responses related to recombinant protein production. Microb Cell Fact 18: 80. doi: 10.1186/s12934-019-1128-7
![]() |
[116] |
Thakur KG, Jaiswal RK, Shukla JK, et al. (2010) Over-expression and purification strategies for recombinant multi-protein oligomers: A case study of Mycobacterium tuberculosis σ/anti-σ factor protein complexes. Protein Express Purif 74: 223-230. doi: 10.1016/j.pep.2010.06.018
![]() |
[117] |
Dzivenu OK, Park HH, Wu H (2004) General co-expression vectors for the overexpression of heterodimeric protein complexes in Escherichia coli. Protein Express Purif 38: 1-8. doi: 10.1016/j.pep.2004.07.016
![]() |
[118] |
Johnston K, Clements A, Venkataramani RN, et al. (2000) Coexpression of proteins in bacteria using T7-based expression plasmids: expression of heteromeric cell-cycle and transcriptional regulatory complexes. Protein Express Purif 20: 435-443. doi: 10.1006/prep.2000.1313
![]() |
[119] |
Rucker P, Torti FM, Torti SV (1997) Recombinant ferritin: modulation of subunit stoichiometry in bacterial expression systems. Protein Eng 10: 967-973. doi: 10.1093/protein/10.8.967
![]() |
[120] |
Kim KJ, Kim HE, Lee KH, et al. (2004) Two-promoter vector is highly efficient for overproduction of protein complexes. Protein Sci 13: 1698-1703. doi: 10.1110/ps.04644504
![]() |
[121] |
McNally EM, Goodwin EB, Spudich JA, et al. (1988) Coexpression and assembly of myosin heavy chain and myosin light chain in Escherichia coli. Proc Natl Acad Sci 85: 7270-7273. doi: 10.1073/pnas.85.19.7270
![]() |
[122] |
Öztürk S, Ergün BG, Çalık P (2017) Double promoter expression systems for recombinant protein production by industrial microorganisms. Appl Microbiol Biotechnol 101: 7459-7475. doi: 10.1007/s00253-017-8487-y
![]() |
[123] |
Ray MVL, Meenan CP, Consalvo AP, et al. (2002) Production of salmon calcitonin by direct expression of a glycine-extended precursor in Escherichia coli. Protein Express Purif 26: 249-259. doi: 10.1016/S1046-5928(02)00523-5
![]() |
[124] | Tao X, Zhao M, Zhang Y, et al. (2019) Comparison of the expression of phospholipase D from Streptomyces halstedii in different hosts and its over-expression in Streptomyces lividans. FEMS Microbiol Lett 366: fnz051. |
[125] |
Liu Y, Shi C, Li D, et al. (2019) Engineering a highly efficient expression system to produce BcaPRO protease in Bacillus subtilis by an optimized promoter and signal peptide. Int J Biol Macromol 138: 903-911. doi: 10.1016/j.ijbiomac.2019.07.175
![]() |
[126] |
Guan C, Cui W, Cheng J, et al. (2016) Construction of a highly active secretory expression system via an engineered dual promoter and a highly efficient signal peptide in Bacillus subtilis. New Biotechnol 33: 372-379. doi: 10.1016/j.nbt.2016.01.005
![]() |
[127] |
Zhang K, Su L, Duan X, et al. (2017) High-level extracellular protein production in Bacillus subtilis using an optimized dual-promoter expression system. Microb Cell Fact 16: 32. doi: 10.1186/s12934-017-0649-1
![]() |
[128] |
Liu X, Wang H, Wang B, et al. (2018) Efficient production of extracellular pullulanase in Bacillus subtilis ATCC6051 using the host strain construction and promoter optimization expression system. Microb Cell Fact 17: 163. doi: 10.1186/s12934-018-1011-y
![]() |
[129] |
Bayat H, Hossienzadeh S, Pourmaleki E, et al. (2018) Evaluation of different vector design strategies for the expression of recombinant monoclonal antibody in CHO cells. Prep Biochem Biotechnol 48: 160-164. doi: 10.1080/10826068.2017.1421966
![]() |
[130] |
Lueking A, Holz C, Gotthold C, et al. (2000) A system for dual protein expression in Pichia pastoris and Escherichia coli. Protein Express Purif 20: 372-378. doi: 10.1006/prep.2000.1317
![]() |
[131] |
Sinah N, Williams CA, Piper RC, et al. (2012) A set of dual promoter vectors for high throughput cloning, screening, and protein expression in eukaryotic and prokaryotic systems from a single plasmid. BMC Biotechnol 12: 54. doi: 10.1186/1472-6750-12-54
![]() |
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