# Determinantal Representations of the Weighted Core-EP, DMP, MPD, and CMP Inverses

Authors: Ivan I. Kyrchei
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, new notions of the weighted core-EP left inverse and the weighted MPD inverse which are dual to the weighted core-EP (right) inverse and the weighted DMP inverse, respectively, are introduced and represented. The direct methods of computing the weighted right and left core-EP, DMP, MPD, and CMP inverses by obtaining their determinantal representations are given. A numerical example to illustrate the main result is given.


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## Body

## 1. Introduction

In the whole article,ℝ and ℂ stand for fields of the real and complex numbers, respectively. ℂm×n and ℂrm×n are reserved for the set of all m×n matrices over ℂ and its subset of matrices with a rank r. For A∈ℂm×n, the symbols ΑT, A∗, and rkA specify the transpose, the conjugate transpose, and the rank of A, respectively. A or detA denotes its determinant. A matrix A∈ℂn×n is Hermitian if A∗=A. The index of A∈ℂn×n, denoted IndA=k, is the smallest positive number such that rkAk+1=rkAk.Definition 1 (see [1]).
ForA∈ℂm×n and W∈ℂn×m, the W-weighted Drazin inverse of A with respect to W, denoted by Ad,W, is the unique solution to equations:(1)AWk+1XW=AWk,XWAWX=X,AWX=XWA,where k=IndAW.The properties of the W-weighted Drazin inverse and some of its applications can be found in [2–8]. Among them, if A∈ℂm×n with respect to W∈ℂn×m, then(2)Ad,W=AWAd2=AWd2A.LetA∈ℂn×n and W=In be the identity matrix of order n. Then, X=Ad is the Drazin inverse of A. In particular, if IndA=1, then the matrix X is called the group inverse and it is denoted by X=A#.Definition 2 (see [9]).
The Moore–Penrose inverse ofA∈ℂn×m is called the exclusive matrix X, denoted by A†, satisfying the following four equations:(3)AXA=A,XAX=X,AX∗=AX,XA∗=XA.For an arbitrary matrixA∈ℂm×n, it is denoted by(i)
NA=x∈ℂn×1:Ax=0, the kernel (or the null space) of A(ii)
CA=y∈ℂm×1:y=Ax,x∈ℂn×1, the column space (or the range space) of A(iii)
ℛA=y∈ℂ1×n:y=xA,x∈ℍ1×m, the row space of APA:=AA† and QA:=A†A are the orthogonal projectors onto the range of A and the range of A∗, respectively.For the recent important results regarding generalized inverses see [10, 11].The core inverse was introduced by Baksalary and Trenkler in [12]. Later, it was investigated by Malik and Thome in [13] and Xu et al. in [14], among others.Definition 3 (see [12]).
A matrixX∈ℂn×n is called the core inverse of A∈ℂn×n if it satisfies the conditions:(4)AX=PA,CX=CA.
When such matrixX exists, it is denoted by A#.The core inverse was extended to the core-EP inverse defined by Prasad and Mohana [15]. Based on the determinantal representation of an reflexive inverse [16, 17], determinantal formulas for the core-EP inverse have been derived in [15].Other generalizations of the core inverse were introduced, namely, BT inverses [18], DMP inverses [13], and CMP inverses [19]. The characterizations, computing methods, some applications of the core inverse, and its generalizations were investigated (see, e.g., [20–30]). Recently, determinantal representations of the core inverse and its generalizations have been obtained in both cases for matrices over the field of complex numbers [31] by using usual determinants and over the quaternion skew field [32] by using noncommutative row-column determinants introduced in [33, 34].Only recent generalizations of the core inverse were extended to rectangular matrices by using the W-weighted Drazin inverse. Among them, the weighted core-EP inverse was introduced in [35], its representations and properties were studied in [36, 37], and generalizations of the weighted core-EP inverse were expanded over a ring with involution [38] and Hilbert space [39], respectively. The concepts of the complex weighted DMP and CMP inverses were introduced and explored in [40, 41], respectively.The main goals of this paper are to introduce and represent new notions of the weighted core-EP left inverse and the weighted MPD inverse which are dual to the weighted core-EP (right) inverse and the weighted DMP inverse, respectively. Also, the direct methods of computing the weighted right and left core-EP, DMP, MPD, and CMP inverses by obtaining their determinantal representations are given.The determinantal representation of the usual inverse is the matrix with cofactors in entries that suggests a direct method of finding the inverse of a matrix. The same is desirable for the generalized inverses. But, there are various expressions of determinantal representations of generalized inverses which are in regard to the search of their more applicable explicit expressions (see, e.g., [16, 17, 42–44]).In this paper, determinantal representations of a generalized inverse obtained based on their limit representations are used, namely, for the Moore–Penrose inverse in [44, 45], for the Drazin inverse in [46, 47], and for the W-weighted Drazin inverse in [48–50].The paper is organized as follows. Section2 starts with a preliminary introduction of the determinantal representations of the Moore–Penrose inverse, of the Drazin and weighted Drazin inverses, and of the core inverse and its generalizations. Section 3 introduces the concepts of the left weighted core-EP inverse and gives determinantal representations of both left and right weighted core-EP inverse. In Section 4, the weighted DMP and MPD inverses are established and determinantal representations of the weighted DMP and MPD inverses are obtained. Determinantal representations of the CMP inverse are obtained in Section 5. A numerical example to illustrate the main results is considered in Section 6. Finally, in Section 7, the conclusions are drawn.


## 2. Preliminaries

The following notations for determinantal representations of generalized inverses are used.Letα:=α1,…,αk⊆1,…,m and β≔β1,…,βk⊆1,…,n be subsets with 1≤k≤minm,n. By Aβα, denote a submatrix of A∈ℂm×n with rows and columns indexed by α and β, respectively. Then, Aαα is a principal submatrix of A with rows and columns indexed by α and Aαα is the corresponding principal minor of the determinant A. Suppose that(5)Lk,n:=α:α=α1,…,αk,1≤α1<⋯<αk≤n,stands for the collection of strictly increasing sequences of 1≤k≤n integers chosen from 1,…,n. For fixed i∈α and j∈β, put(6)Ir,mi:=α:α∈Lr,m,i∈α,Jr,nj:=β:β∈Lr,n,j∈β.a.j and a.j∗, ai. and ai.∗ denote the j-th columns and the i-th rows of A and A∗, respectively. By Ai.b and A.jc, the matrices obtained from A by replacing its i-th row with row b and its j-th column with column c are denoted.Lemma 1 (see [44]).
IfA∈ℂrm×n, then the Moore–Penrose inverse A†=aij†∈ℂn×m possesses the determinantal representations:(7)aij†=∑β∈Jr,niA∗A.ia.j∗ββ∑β∈Jr,nA∗Aββ=∑α∈Ir,mjAA∗j.ai.∗αα∑α∈Ir,mAA∗αα.Remark 1.
For an arbitrary full-rank matrixA∈ℂrm×n, a row-vector b∈ℂ1×m, and a column-vector c∈ℂn×1, it is given as, respectively,(8)AA∗i.b=∑α∈Im,miAA∗i.bαα,i=1,…,m,AA∗=∑α∈Im,mAA∗αα,whenr=m;A∗A.jc=∑β∈Jn,njA∗A.jcββ,j=1,…,n,A∗A=∑β∈Jn,nA∗Aββ,whenr=n.Corollary 1 (see [45]).
LetA∈ℂrm×n. Then, the following determinantal representations can be obtained:(i)
For the projectorQA=qijn×n,(9)qij=∑β∈Jr,niA∗A.ia˙.jββ∑β∈Jr,nA∗Aββ=∑α∈Ir,njA∗Aj.a˙i.αα∑α∈Ir,nA∗Aαα,wherea˙.j is the j-th column and a˙i. is the i-th row of A∗A;(ii)
For the projectorPA=pijm×m,(10)pij=∑α∈Ir,mjAA∗j.a¨i.αα∑α∈Ir,mAA∗αα=∑β∈Jr,miAA∗.ia¨.jββ∑β∈Jr,mAA∗ββ,wherea¨i. is the i-th row and a¨.j is the j-th column of AA∗.Lemma 2 (see [48]).
LetA∈ℂm×n, W∈ℂn×m, and k=maxIndAW,IndWA. Then, Ad,W=aijd,W∈ℂm×n can be expressed as(11)aijd,W=∑β∈Jr,miAW.ik+2v¯.jββ∑β∈Jr,mAWk+2ββ(12)=∑α∈Ir,njWAj.k+2u¯i.αα∑α∈Ir,nWAk+2αα,where u¯i. is the i-th row of U¯=AWAk and v¯.j is the j-th column of V¯=AWkA.The two core inverses are introduced.Definition 4 (see [12]).
A matrixX∈ℂn×n is said to be the (right) core inverse of A∈ℂn×n if it satisfies the conditions:(13)AX=PA,CX=CA.
When such matrixX exists, it is denoted by A.Definition 5.
(see [31]). A matrix X∈ℂn×n is said to be the left core inverse of A∈ℂn×n if it satisfies the conditions:(14)XA=QA,ℛX=ℛA.
When such matrixX exists, it is denoted by A.Similar as in [15], two core-EP inverses are introduced.Definition 6 (see [15]).
A matrixX∈ℂn×n is said to be the right core-EP inverse of A∈ℂn×n if it satisfies the conditions:(15)XAX=A,CX=CX∗=CAd.
It is denoted byA†.The following lemma gives characterization of the right core-EP inverse.Lemma 3 (see [15]).
LetA,X∈ℂn×n be such that IndA=k. Then, X is the right core-EP inverse of A if and only if X satisfies the conditions:(16)XAk+1=Ak,AX2=X,AX∗=AX,CX⊆CAk.Lemma 4 (see [22], Theorem 2.3).
LetA∈ℂn×n and let l be a nonnegative integer such that l≥k=IndA. Then, A=AdAlAl†.Definition 7.
A matrixX∈ℂn×n is said to be the left core-EP inverse of A∈ℂn×n if it satisfies the conditions:(17)XAX=A,ℛX=ℛX∗=ℛAd.
It is denoted byA†.Remark 2.
SinceCA∗d=ℛAd, then the left core inverse A of A∈ℂn×n is similar to the ∗core inverse introduced in [15] and the dual core-EP inverse introduced in [30].Similarly, the following characterization of the left core-EP inverse is obtained.Lemma 5 (see [30]).
LetX,A∈ℂn×n and let l be a nonnegative integer such that l≥k=IndA. The following statements are equivalent:(i)
X is the left core-EP inverse of A(ii)
Ak+1X=Ak,X2A=X,XA∗=XA,andℛX⊆ℛAk(iii)
X=A=Al†AlAdThanks to [22], there exists the simple relation between the left and right core-EP inverses, A∗=A∗○†. So, it is enough to investigate the left core-EP inverse, so the right core-EP inverse case can be investigated analogously. But in [31], separate determinantal representations of both core-EP inverses and the both core inverses are given.Lemma 6 (see [31]).
SupposeA∈ℂn×n, IndA=k, rkAk=s, and there exist A○†=aij○†,r and A○†=aij○†,l. Then, they have the following determinantal representations, respectively,(18)aij○†,r=∑α∈Is,njAk+1Ak+1∗j.a^i.αα∑α∈Is,nAk+1Ak+1∗αα,(19)aij†,l=∑β∈Js,niAk+1∗Ak+1.iaˇ.jββ∑β∈Js,nAk+1∗Ak+1ββ,where a^i. is the i-th row of A^=AkAk+1∗ and aˇ.j is the j-th column of Aˇ=Ak+1∗Ak.
IfIndA=1, then A○#=aij○#,r and A○#=aij○#,l have the following determinantal representations, respectively,(20)aij○#,r=∑α∈Is,njA2A2∗j.a^i.αα∑α∈Is,nA2A2∗αα,aij○#,l=∑β∈Js,niA2∗A2.iaˇ.jββ∑β∈Js,nA2∗A2ββ,where a^i. is the i-th row of A^=AA2∗ and aˇ.j is the j-th column of Aˇ=A2∗A.


## 3. Concepts of the W-Weighted Core-EP Inverses and Their Determinantal Representations

The concept of the W-weighted core-EP inverse was introduced by Ferreyra et al. in [35].Definition 8.
SupposeA∈ℂm×n, W∈ℂn×m, and k=maxIndWA,IndAW, then the (right) W-weighted core-EP inverse of A is the unique solution to the system:(21)WAWX=WAkWAk†,CX⊆CAWk.
It is denoted byA○†,W,r.From [35], the right weighted core-EP inverse can be determined as follows.Lemma 7.
(see [35]). Let A,X∈ℂm×n, W∈ℂn×m, and k=maxIndWA,IndAW. The following statements are equivalent:(i)
X is the right weighted core-EP inverse of A.(ii)
XWAWk+1=AWk,AWXWX=X,andWAWX∗=WAWX.(22)X=AWA○†2.Introduction of a left weighted core-EP inverse is proposed as well.Definition 9.
SupposeA∈ℂm×n, W∈ℂn×m, and k=maxIndWA,IndAW. The left W-weighted core-EP inverse of A is the unique solution to the system:(23)XWAW=AWk†AWk,ℛX⊆ℛWAk.
It is denoted byA○†,W,l.Theorem 1.
LetA,X∈ℂm×n, W∈ℂn×m, and k=maxIndWA,IndAW. The following statements are equivalent:(i)
X=AW○†2A.(ii)
X is the left weighted core-EP inverse of A.(iii)
X is the unique solution to the three equations:(24)WAk+1WX=WAk,XWXWA=X,XWAW∗=XWAW.Proof.
i⟼ii. It is shown that X=AW○†2A satisfies condition (23). Indeed,(25)XWAW=AW○†2AWAW=AW○†AW=AWk†AWkAWdAW=AWk†AWk,AW○†2A=AW○†3AWA=⋯=AW○†k+2AWkA=AW○†k+2AWAk,i.e.,ℛAW○†2A⊆ℛWAk.
ii⟼iii. Now, it is needed to prove that X=AW○†2A satisfies the equations in (24).
SinceWAk+1W=WAWk+1 and from Lemma 5,(26)AW○†=AWk†AWkAWd,then(27)WAk+1WAW○†2A=WAWk+1AW○†WA○†A=WAWkAWk†AWkAWdA=WAWkAWdA=WAk+1WAd=WAk.
By Lemma5, taking into account AW○†2AW=AW○†, it is obtained that(28)AW○†2AWAW○†2AWA=AW○†2AWAW○†A=AW○†2A.
Finally,(29)AW○†2AWAW∗=AW○†AW∗=AW○†AW.
Now, the uniqueness ofX is proven. Let(30)WAk+1WX=WAk,XWXWA=X,XWAW∗=XWAW,X=AW○†2A.
Suppose there also exists the left weighted core-EP inverseY such that(31)WAk+1WY=WAk,YWYWA=Y,YWAW∗=YWAW.
It is shown thatY=X=AW○†2A. So,(32)Y=YWYWA=YWY2WA2=YWYkWAk=YWYkWAk+1WX=YWk+1AWk+1X=YWAWk+1X=AWk†AWkk+1X=AW2k+1†AW2k+1AW○†2A.
By Lemma5, AW○†=AW2k+1†AW2k+1AWd. So,(33)Y=AW2k+1†AW2k+1AW2k+1†AW2k+1AWdAW○†A=AW2k+1†AW2k+1AWdAW○†A=AW○†2A.
Finally, from the uniqueness ofX follows iii↦i.Now, determinantal representations of the W-weighted core-EP inverses are given.Theorem 2.
SupposeA∈ℂm×n, W∈ℂn×m, and k=maxIndWA,IndAW, rkWAk=s. Denoting WA≔U=uij∈ℂn×n. Then, the right weighted core-EP inverse A○†,W,r=aij○†,W,r∈ℂm×n possesses the determinantal representations:(34)aij○†,W,r=∑α∈Is,njUk+1Uk+1∗j.ϕ˜i.αα∑α∈Is,nUk+1Uk+1∗αα2,where ϕ˜i. is the i-th row of Φ˜=ΦUkUk+1∗. The matrix Φ=ϕif is determined as follows:(35)ϕif=∑α∈Is,nfUk+1Uk+1∗f.u˜i.αα,where u˜i. is the i-th row of U˜=AUkUk+1∗.Proof.
Taking into account (22) for A○†,W,r, it is obtained that(36)aij○†,W,r=∑l=1n∑f=1nailulf○†,rufj○†,r.
Using the determinantal representation (18) of U○† gives(37)ϕif=∑l=1nailulf○†,r=∑l=1nail∑α∈Is,nfUk+1Uk+1∗f.u^l.αα∑α∈Is,nUk+1Uk+1∗αα=∑α∈Is,nfUk+1Uk+1∗f.u˜i.αα∑α∈Is,nUk+1Uk+1∗αα,where u^l. is the l-th row of U^=UkUk+1∗ and u˜i. is the i-th row of U˜=AUkUk+1∗. Denoting(38)ϕif=∑α∈Is,nfUk+1Uk+1∗f.u˜i.αα.
Substitutingϕif into (36) gives(39)aij○†,W,r=∑l=1nϕifufj○†,r=∑f=1nϕif∑α∈Is,nfUk+1Uk+1∗j.u^f.αα∑α∈Is,nUk+1Uk+1∗αα2.
Putting∑f=1nϕifu^f.=ϕ˜i. as the i-th row of Φ˜=ΦUkUk+1∗ yields (34).Similarly, theorem on the determinantal representation of the left W-weighted core-EP inverse can be proved.Theorem 3.
SupposeA∈ℂm×n, W∈ℂn×m, and k=maxIndWAIndAW, rkAWk=s. Denoting AW:=V=vij∈ℂm×m. Then, the left W-weighted core-EP inverse A†,W,l=aij†,W,l∈ℂn×m possesses the determinantal representations(40)aij†,W,l=∑β∈Js,miVk+1∗Vk+1.iψ˜.jββ∑β∈Js,mVk+1∗Vk+1ββ2,where ψ˜.j is the j-th column of Ψ˜=Vk+1∗VkΨ. The matrix is determined, Ψ=ψlj∈ℂm×n, as follows:(41)ψlj=∑β∈Js,mlVk+1∗Vk+1.lv˜.jββ,where v˜.j is the j-th column of V˜=Vk+1∗VkA.


## 4. The W-Weighted DMP and MPD Inverses and Their Determinantal Representations

The concept of the DMP inverse was introduced by Malik and Thome as follows.Definition 10 (see [13]).
Suppose thatA∈ℂn×n and IndA=k. A matrix X∈ℂn×n is said to be the DMP inverse of A if it satisfies the conditions:(42)XAX=X,XA=AdA,AkX=AkA†.
It is denoted byAd,†.It is proved in [13] that if a matrix satisfies the system of equations (42), then it is unique and has the representation Ad,†=AdAA†.In [31], the MPD inverse is introduced.Definition 11.
SupposeA∈ℂn×n and IndA=k. A matrix X∈ℂn×n is said to be the MPD inverse of A if it satisfies the conditions:(43)XAX=X,AX=AAd,XAk=A†Ak.
It is denoted byA†,d.It is not difficult to show thatA†,d is unique and it can be represented as A†,d=A†AAd.In [31], the determinantal representations of the DMP and MPD inverses are given.Recently in [40], the definition of the DMP inverse of a square matrix was extended to rectangular matrices.Definition 12 (see [40]).
LetA∈ℂm×n and W∈ℂn×m be a nonzero matrix. The W-weighted DMP (WDMP) inverse of A with respect to W is defined as(44)Ad,†,W=WAd,WWAA†.Lemma 8 (see [40]).
LetA∈ℂm×n, W∈ℂn×m be a nonzero matrix, and k=maxIndWAIndAW. The matrix X=Ad,†,W is the unique matrix that satisfies the following system of equations:(45)XAX=X,XA=WAd,WWA,WAk+1X=WAk+1A†.Introduction of the weighted MPD inverse is proposed as well.Theorem 4.
LetA∈ℂm×n, W∈ℂn×m be a nonzero matrix, and k=maxIndWA,IndAW. Then, the matrix X=A†AWAd,WW is the unique solution to the equations:(46)XAX=X,AX=AWAd,WW,XAWk+1=A†AWk+1.Proof.
From Definitions1 and 2 and taking into account (2), it follows(47)A†AWAd,WWAA†AWAd,WW=A†AWAd,WWAWAd,WW=A†AWAd,WW,AA†AWAd,WW=AWAd,WW,A†AWAd,WWAWk+1=A†AWAWd2AWAWk+1=A†AWAWdAWAWdAWk+1=A†AWk+1.
It means thatX=A†AWAd,WW is the solution to equation (46).
To prove uniqueness, suppose bothX1 and X2 are two solutions to (46). Using repeated applications of the equations in (46) and in Definition 1, it is obtained that(48)X1=X1AX1=X1AWAd,WW=X1AW2Ad,WW2=⋯=X1AWk+1Ad,WWk+1=A†AWk+1Ad,WWk+1=X2AWk+1Ad,WWk+1=X2AWAd,WW=X2AX2=X2.
It completes the proof.Definition 13.
LetA∈ℂm×n and W∈ℂn×m be a nonzero matrix. The W-weighted MPD (WMPD) inverse of A with respect to W is defined as(49)A†,d,W=A†AWAd,WW.Now, determinantal representations of the WDMP inverse are given.Theorem 5.
LetA∈ℂrm×n and W∈ℂn×m be a nonzero matrix. Suppose k=maxIndWA,IndAW, and rkWAk=rkUk=r1. Then, the determinantal representations of its WDMP inverse Ad,†,W=aijd,†,W can be expressed as(50)aijd,†,W=∑α∈Ir,mjAA∗j.ω˜i.αα∑α∈Ir,mAA∗ααWAk+2αα,where ω˜i. is the i-th row of Ω˜=ΩWAA∗. The matrix Ω=ωis is such that(51)ωis:=∑α∈Ir1,nsWAs.k+2u^i.αα,where u^i. is the i-th row of U^=WAk+1.Proof.
Taking into account (42), the following is obtained:(52)aijd,†,W=∑t=1m∑s=1n∑f=1mwitatsd,WwsfpfjA,where Ad,W=atsd,W∈ℂm×n and PA=pfjA∈ℂm×m. Applying the determinantal representations (12) of Ad,W and (10) of PA in (52) gives(53)aijd,†,W=∑s=1n∑α∈Ir1,nsWAs.k+2u^i.αα∑α∈Ir,mjAA∗j.ws.2αα∑α∈Ir1,nWAk+2αα∑α∈Ir,mAA∗αα,where u^i. is the i-th row of U^:=WAk+1∈ℍn×n and ws.2 is the s-th row of W2=WAA∗.
Denote(54)ωis:=∑α∈Ir1,nsWAs.k+2u^i.αα,and construct the matrix Ω=ωis. Since(55)∑s=1nωis∑α∈Ir,mjAA∗j.ws.2αα=∑α∈Ir,mjAA∗j.ω˜i.αα,where ω˜i. is the i-th row of Ω˜=ΩW2=ΩWAk+1A∗, then, finally, from (53), it follows (50).It is obtained similarly for the weighted MPD inverse.Theorem 6.
LetA∈ℂrm×n and W∈ℂn×m be a nonzero matrix. Suppose k=maxIndWA,IndAW, and rkAWk=rkVk=r1. Then, the determinantal representations of its WMPD inverse A†,d,W=aij†,d,W can be expressed as follows:(56)aij†,d,W=∑β∈Jr,niA∗A.iυ˜.jββ∑β∈Jr,nA∗Aββ∑β∈Jr1,mAWk+2ββ2,where υ˜.j is the j-th column of ϒ˜=A∗AWϒ. The matrix ϒ=υsj is determined by(57)υsj=∑β∈Jr1,msAW.sk+2v˜.jββ,where v˜.j is the j-th column of V˜=AWk+1.Proof.
Taking into account (45), the following is obtained:(58)aij†,d,W=∑t=1m∑s=1n∑f=1mqifAwftatsd,Wwsf,where Ad,W=atsd,W∈ℂm×n and QA=qifA∈ℂn×n. Applying the determinantal representations (11) of Ad,W and (9) of QA in (58) gives(59)aij†,d,W=∑t=1m∑β∈Jr,niA∗A.iw.t2ββ∑β∈Jr,nA∗Aββ∑β∈Jr1,mtAW.tk+2v˜.jββ∑β∈Jr1,mAWk+2ββ,where w.t2 is the t-th column of W2=A∗AW∈ℂm×m and v˜.j is the j-th column of V˜=AWk+1. Now, construct the matrix ϒ=υtj, where(60)υtj:=∑β∈Jr1,mtAW.tk+2v˜.jββ.
Then, from (59) it follows (56).Theorems5 and 6 give the determinantal representations of the weighted DMP and MPD inverses. For better understanding, the algorithm of finding one of them, for example, the WDMP inverse from Theorem 5, is presented.Algorithm 1.
(1)
Compute the matrixUˇ=Uk+1=WAk+1.(2)
By (51), find ωis for all i,s=1,…,n and construct the matrix Ω=ωis.(3)
Compute the matrixΩ˜≔ΩWAA∗.(4)
Finally, findaijd,†,W by (50) for all i=1,…,m and j=1,…,n.


## 5. Determinantal Representations of the Weighted CMP Inverse

In [19], Mehdipour and Salemi investigated the CMP inverse.Definition 14 (see [19]).
SupposeA∈ℂn×n has the core-nilpotent decomposition A=A1+A2, where IndA1=IndA, A2 is nilpotent, and A1A2=A2A1=0. The CMP inverse of A is called the matrix Ac,†:=A†A1A†.Lemma 9 (see [19]).
LetA∈ℂn×n. The matrix X=Ac,† is the unique matrix that satisfies the following system of equations:(61)XAX=X,AXA=A1,AX=A1A†,XA=A†A1.
Moreover,Ac,†=A†AAdAA†.Determinantal representations of the CMP inverse are derived in [31].Recently, Mosić [41] introduced the weighted CMP inverse of a rectangular matrix.Lemma 10 (see [41]).
LetA∈ℂm×n and W∈ℂn×m be a nonzero matrix. The system of equations(62)XAX=X,AX=AWAd,WWAA†,XA=A†AWAd,WWA,is consistent, and its unique solution is X=A†AWAd,WWAA†.Definition 15.
LetA∈ℂm×n and W∈ℂn×m be a nonzero matrix. The weighted CMP (WCMP) inverse of A with respect to W is defined as(63)Ac,†,W=A†AWAd,WWAA†.Theorem 7.
LetA∈ℂrm×n and W∈ℂn×m be a nonzero matrix. Suppose k=maxIndWA,IndAW. Then, the determinantal representations of its WCMP inverse Ac,†,W=aijc,†,W can be expressed as follows:(i)
IfrkWAk=r1, then(64)aijc,†,W=∑α∈Ir,mjAA∗j.ω˜i.αα∑β∈Jr,nA∗Aββ2∑α∈Ir1,nWAk+2αα,whereω˜i. is the i-th row of Ω˜=ΩWAA∗. The matrix Ω=ωis is such that(65)ωis=∑α∈Ir1,nsWAs.k+2ϕi.1αα.Here,ϕi.1 is the i-th row of Φ1=ΦAWAk and the matrix Φ=ϕit is such that(66)ϕit:=∑β∈Jr,niA∗A.iw.t1ββ,wherew.t1 is the t-th column of W1=A∗AW.(ii)
IfrkAWk=r1, then(67)aijc,†,W=∑β∈Jr,niA∗A.iυ˜.jββ∑α∈Ir,mAA∗αα2∑β∈Jr1,mAWk+2ββ,whereυ˜.j is the j-th column of ϒ˜=A∗AWϒ. The matrix ϒ=υtj is determined by(68)υtj=∑β∈Jr1,mtAW.tk+2ψ.j1ββ,whereψ.j1 is the j-th column of Ψ1=AWkAΨ. Here, Ψ=ψsj is such that(69)ψsj:=∑α∈Ir,mjAA∗j.ws.2αα,wherews.2 is the s-th row of W2=WAA∗.Proof.
(i)
Taking into account (63) and applying one of the cases of (9) and (10) for the determinantal representations of QA and PA, respectively, and (12) for the determinantal representation of Ad,W give(70)aijc,†,W=∑t=1m∑s=1n∑β∈Jr,niA∗A.iw.t1ββ∑β∈Jr,nA∗Aββ×∑α∈Ir1,nsWAs.k+2u¯t.αα∑α∈Ir1,nWAk+2αα∑α∈Ir,mjAA∗j.ws.2αα∑α∈Ir,mAA∗αα,wherew.t1 is the t-th column of W1=A∗AW, u¯t. is the t-th row of U¯=AWAk, and ws.2 is the s-th row of W2=WAA∗.Denote(71)ϕit:=∑β∈Jr,niA∗A.iw.t1ββ,and construct the matrixΦ=ϕit. Then, determine(72)ωis=∑t=1nϕit∑α∈Ir1,nsWAs.k+2u¯t.αα=∑α∈Ir1,nsWAs.k+2ϕi.1αα,whereϕi.1 is the i-th row of Φ1=ΦAWAk and construct the matrix Ω=ωis. Taking into account that ∑α∈Ir,mAA∗αα=∑β∈Jr,nA∗Aββ, and(73)∑s=1nωis∑α∈Ir,mjAA∗j.ws.2αα=∑α∈Ir,mjAA∗j.ω˜i.αα,whereω˜i. is the i-th row of Ω˜=ΩW2=ΩWAA∗, finally from (70), it follows (64).(ii)
By applying (11) for the determinantal representation of Ad,W and the same determinantal representations of QA and PA as in the point (i), it is obtained that(74)aijc,†,W=∑t=1m∑s=1n∑β∈Jr,niA∗A.iw.t1ββ∑β∈Jr,nA∗Aββ×∑β∈Jr1,mtAW.tk+2v¯.sββ∑β∈Jr1,mAWk+2ββ∑α∈Ir,mjAA∗j.ws.2αα∑α∈Ir,mAA∗αα,where w.t1 is the t-th column of W1:=A∗AW, v¯.s is the s-th column of V¯=AWkA, and ws.2 is the s-th row of W2≔WAA∗. Denote(75)ψsj:=∑α∈Ir,mjAA∗j.ws.2αα,and construct the matrix Ψ=ψsj. Then, introduce(76)υtj=∑s=1n∑β∈Jr1,mtAW.tk+2v¯.sββψsj=∑β∈Jr1,mtAW.tk+2ψ.j1ββ,where ψ.j1 is the j-th column of Ψ1=AWkAΨ, and construct the matrix ϒ=υtj. Taking into account that(77)∑t=1n∑β∈Jr,niA∗A.iw.t1ββυtj=∑β∈Jr,niA∗A.iυ˜.jββ,where υ˜.j is the j-th column of ϒ˜=A∗AWϒ, finally from (74), it follows (67).Theorem7 gives determinantal representations of the WCMP inverse. For better understanding, the algorithm of its finding, for example, in Theorem 7 with the Case (i), is presented The algorithm by Theorem 7 with Case (ii) can be constructed similarly.Algorithm 2.
(1)
Compute the matrixW1=A∗AW.(2)
Findϕit by (66) for all i,t=1,…,n and construct the matrix Φ=ϕit.(3)
Compute the matrixΦ1=ΦAWAk.(4)
By (65), find ωis for all i=1,…,m and s=1,…,n, and construct the matrix Ω=ωis.(5)
Compute the matrixΩ˜=ΩWAA∗.(6)
Finally, findaijc,†,W by (64) for all i=1,…,m and j=1,…,n.


## 6. An Example

In this section, an example is given to illustrate the results. Given the matrices(78)A=0iii0i1−10112,W=i010−1i01i−1i11.Since(79)V=AW=−1−2i−2i2i−2−i−12ii1+i−i1−1−3+3i3i33,U=WA=1−2−101−i1−i1−1−i−i,rkV=rkV2=2, and rkU2=rkU=2, then k=maxIndAW,IndWA=1.The weighted DMP inverse is obtained from Algorithm1.(1)
Computing the matrixU^=U2 and AA∗,(80)U2=0−3+3i−3+3i1−i−2−2i−1−3i1−i−5+i−4,U3=−3+3i6+6i3+9i−4i−8+6i−8+2i−3−i−2+12i−5+11i.(2)
By (51), ωis for all i,s=1,…,n is found. So,(81)ωis:=∑α∈I2,31WA1.3u^1.αα=0−3+3i−4i−8+6i+0−3+3i−3−i−5+11i=−24−6i.Continuing similarly, it is obtained that(82)Ω=−24−6i12−24i−12−30i12+12i−6−6i6+6i−12+6i6−30i−6−24i.(3)
Compute the matrix(83)Ω˜=ΩWAA∗=0−54+54i−54−54i−54−54i−54+54i054+54i−54−54i−54+54i−54+54i0−108−108i,AA∗=21−i3i12i3ii−i20−3i3i06,and the values(84)∑β∈J2,4AA∗ββ=27,∑α∈I,nWA3αα=−54−54i.(4)
Finally,aijd,†,W by (50) for all i=1,…,4 and j=1,2,3 is found. So,(85)a11d,†,W=∑α∈I2,41rdet1AA∗1.ω˜1.αα∑β∈I2,4AA∗αα∑α∈I2,3WA3αα=127−54−54i0−54+54i12+0−54+54ii2+0−54−54i−3i6=i9.Continuing similarly, it is obtained that(86)Ad,†,W=19i−2i31−2ii−31−i−i02.It is easy to verify thatX=Ad,†,W from (86) with the given matrices (86) is the solution to equation (45).Similarly, the WMPD inverse can be found:(87)A†,d,W=16−3+i−3+3i43+3i−2i0−20−3−i−3+3i23+3i.


## 7. Conclusions

In this paper, new notions of the weighted core-EP left inverse and the weighted MPD inverse that are dual to the weighted core-EP (right) inverse and the weighted DMP inverse, respectively, are introduced and explored. Using their determinantal representations, the direct methods of computing the weighted right and left core-EP, DMP, MPD, and CMP inverses are given.



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*Source: 9816038-2020-05-31.xml*