# Determinantal Representations of the Core Inverse and Its Generalizations with Applications

Authors: Ivan I. Kyrchei
Journal: Journal of Mathematics
 (2019)

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## Abstract

In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equations that is an analog of Cramer’s rule. A numerical example to illustrate the main result is given.


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## Body

## 1. Introduction

In the whole article, the notationsℝ and ℂ are reserved for fields of the real and complex numbers, respectively. ℂm×n stands for the set of all m×n matrices over ℂ. ℂrm×n determines its subset of matrices with rank r. For A∈ℂm×n, the symbols AT, A∗, and rkA specify the transpose, the conjugate transpose, and the rank of A, respectively. A or detA stands for its determinant. A matrix A∈ℂn×n is Hermitian if A∗=A.A
† means the Moore-Penrose inverse of A∈ℂn×m, i.e., the exclusive matrix X satisfying the following four equations:(1)AXA=A,(2)XAX=X,(3)AX∗=AX,(4)XA∗=XA.ForA∈ℂn×n with index IndA=k, i.e., the smallest positive number such that rkAk+1=rkAk, the Drazin inverse of A, denoted by Ad, is called the unique matrix X that satisfies equation (2) and the following equations:(5)AX=XA,(6a)XAk+1=Ak,(6b)Ak+1X=Ak.In particular, ifIndA=1, then the matrix X is called the group inverse and it is denoted by X=A#. If IndA=0, then A is nonsingular and Ad=A†=A−1.It is evident that if the condition (5) is fulfilled, then (6a) and (6b) are equivalent. We put both these conditions because they will be used below independently of each other and without the obligatory fulfillment of (5).A matrixA satisfying the conditions i,j,… is called an i,j,…-inverse of A and is denoted by Ai,j,…. The set of matrices Ai,j,… is denoted by Ai,j,…. In particular, A1 is called the inner inverse of A, A2 is called the outer inverse of A, A1,2 is called the reflexive inverse of A, A1,2,3,4 is its Moore-Penrose inverse, etc.For an arbitrary matrixA∈ℂm×n, we denote 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∈ℂ1×m, the row space of AP
A
≔
A
A
† and QA≔A†A are the orthogonal projectors onto the range of A and the range of A∗, respectively.The core inverse was introduced by Baksalary and Trenkler in [1]. Later, it was investigated by Xu et al. in [2], among others. Rakić et al. in [3] generalized the core inverse of a complex matrix to the case of an element in a ring.Definition 1 [1].
A matrixX∈ℂn×n is called the core inverse of A∈ℂn×n if it satisfies the conditions(7)AX=PA,CX=CA.
When such matrixX exists, it is denoted by A○#.
In 2014, the core inverse was extended to the core-EP inverse defined by Manjunatha Prasad and Mohana [4]. Other generalizations of the core inverse were recently introduced for n×n complex matrices, namely, BT inverses [5], DMP inverses [6], CMP inverses [7], etc. The characterizations, computing methods, some applications of the core inverse, and its generalizations were recently investigated in complex matrices and rings (see, e.g., [3, 8–18]).
In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, for generalized inverse matrices there exist different determinantal representations as a result of the search of their more applicable explicit expressions (see, e.g., [19–25]). In this paper, we get new determinantal representations of the core inverse and its generalizations by using the author’s recently obtained determinantal representations of the Moore-Penrose inverse and the Drazin inverse over the quaternion skew field and the field of complex numbers as a special case [26, 27].
The results concerning quaternion matrices have been achieved thanks to the theory of row-column determinants introduced in [28, 29]. Within the framework of the theory of row-column determinants, determinantal representations of various kind of generalized inverses, generalized inverse solutions (analogs of Cramer’s rule) of quaternion matrix equations have been derived by the author (see, e.g., [30–38]) and by other researchers (see, e.g., [39–41]).
Note that a determinantal representation of the core-EP generalized inverse in complex matrices has been derived in [4] based on the determinantal representation of a reflexive inverse obtained in [19, 20].
One of possible applications of the determinantal representations of the inverse matrix is Cramer’s rule to find solutions of systems of linear equations. Determinantal representations of generalized inverses have similar applications. In this paper, we derive the determinantal representation of the Bott-Duffin inverse that has close relation with the core inverse, and apply it to obtain Cramer’s rule for the constrained linear equations.
The paper is organized as follows. In Section2, we start with preliminary introduction of determinantal representations of the Moore-Penrose inverse and the Drazin inverse. In the next sections, we give determinantal representations of the core inverse and its generalizations. In particular, determinantal representations of the right and left core inverses are established in Section 3, of the right and left core-EP inverses in Section 4, and of the DMP inverse and its dual MPD inverse in Section 5. Determinantal representations of the CMP inverse are obtained in Section 6. In Section 7, we derive Cramer’s rule for the constrained linear equations by using the determinantal representation of the Bott-Duffin inverse that is same as for the right core inverse. A numerical example to illustrate the main results is considered in Section 8. Finally, in Section 9, the conclusions are drawn.


## 2. Preliminaries

Letα≔α1,…,αk⊆1,…,m and β≔β1,…,βk⊆1,…,n be subsets with 1≤k≤minm,n. The submatrix of A∈ℂm×n with rows and columns indexed by α and β, respectively, and denoted by Aβα. 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(8)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 Ir,mi≔α:α∈Lr,m,i∈α and Jr,nj≔β:β∈Lr,n,j∈β.Denote bya.j and a.j∗, ai. and ai.∗ the jth columns and the ith rows of A and A∗, respectively. By Ai.b and A.jc, we denote the matrices obtained from A by replacing its ith row with the row b and its jth column with the column c.Theorem 1 [21].
IfA∈ℂrm×n, then the Moore-Penrose inverse A†=aij†∈ℂn×m possess the determinantal representations(9)aij†=∑β∈Jr,niA∗A.ia.j∗ββ∑β∈Jr,nA∗Aββ,(10)aij†=∑α∈Ir,mjAA∗j.ai.∗αα∑α∈Ir,mAA∗αα.Remark 1.
For an arbitrary full-rank matrixA∈ℂrm×n, a row-vector b∈ℍ1×m, and a column-vector c∈ℍn×1, we mean, respectively,(11)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 [21].
LetA∈ℂrm×n. Then, the following determinantal representations can be obtained:(i)
For the projector QA=qijn×n,(12)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 jth column and a˙i. is the ith row of A∗A.(ii)
For the projectorPA=pijm×m,(13)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 ith row and a¨.j is the jth column of AA∗.The following lemma gives determinantal representations of the Drazin inverse in complex matrices.Lemma 1 [22].
LetA∈ℂn×n with IndA=k and rkAk+1=rkAk=r. Then, the determinantal representations of the Drazin inverse Ad=aijd∈ℂn×n are(14)aijd=∑β∈Jr,niAk+1.ia.jkββ∑β∈Jr,nAk+1ββ=∑α∈Ir,njAk+1j.ai.kαα∑α∈Ir,nAk+1αα,where ai.k is the ith row and a.jk is the jth column of Ak.Corollary 2 [22].
LetA∈ℂn×n with IndA=1 and rkA2=rkA=r. Then, the determinantal representations of the group inverse A#=aij#∈ℂn×n are(15)aij#=∑β∈Jr,niA2.ia.jββ∑β∈Jr,nA2ββ=∑α∈Ir,njA2j.ai.αα∑α∈Ir,nA2αα.


## 3. Determinantal Representations of the Core Inverses

Together with the core inverse in [3], it was introduced the dual core inverse. Since the both these core inverses are equipollent and they are different only in the position relative to the inducting matrix A, we propose to call them as the right and left core inverses regarding to their positions. So, according to [1], we have the following definition that is equivalent to Definition 1.Definition 2.
A matrixX∈ℂn×n is said to be the right core inverse matrix of A∈ℂn×n if it satisfies the conditions(16)AX=PA,CX=CA.
When such matrixX exists, it is denoted by A○#.
The following definition of the left core inverse can be given that is equivalent to the introduced dual core inverse [3].Definition 3.
A matrixX∈ℂn×n is said to be the left core inverse matrix of A∈ℂn×n if it satisfies the conditions(17)XA=QA,ℛX=ℛA.
When such matrixX exists, it is denoted by A○#.Remark 2.
In [42], the conditions of the dual core inverse are given as follows:(18)A○#A=PA∗,CA○#⊆CA∗.
SincePA∗=A∗A∗†=A†A∗=A†A=QA and ℛA=CA∗, then these conditions and (17) are analogous.Remark 3.
In Definitions 2 and 3, we purposely state that these definitions concern with matrices, since the right and core inverses in a ring were introduced recently in [43]. The notions of the right and left core inverse matrices, introduced here, and of the right and left core inverses in a ring, introduced in [43], have no direct connection with each other.
According to [1], we introduce the following sets of matrices:(19)ℂnCM=A∈ℂn×n:rkA2=rkA,ℂnEP=A∈ℂn×n:A†A=AA†=A∈ℂn×n:CA=CA∗.
The matrices fromℂnCM are called group matrices or core matrices. If A∈ℂnEP, then clearly A†=A#. It is known that the core inverses of A∈ℂn×n exist if and only if A∈ℂnCM or IndA=1. Moreover, if A is nonsingular, IndA=0, then its core inverses are the usual inverse. According to [1], we have the following representations of the right and left core inverses.Lemma 2 [1].
LetA∈ℂnCM. Then,(20)A○#=A#AA†,(21)A○#=A†AA#.Remark 4.
In Theorems 2 and 3, we will suppose thatA∈ℂnCM but A∉ℂnEP. if A∈ℂnCM and A∈ℂnEP (in particular, A is Hermitian), then from Lemma 2 and the definitions of the Moore-Penrose and group inverses it follows that A○#=A○#=A#=A†.Theorem 2.
LetA∈ℂnCM and rkA2=rkA=s. Then, its right core inverse matrix A○#=aij○#,r has the following determinantal representations:(22)aij○#,r=∑α∈Is,njAA∗j.ui.1αα∑β∈Js,nA2ββ∑α∈Is,nAA∗αα=∑β∈Js,niA2.iu.j2ββ∑β∈Js,nA2ββ∑α∈Is,nAA∗αα,where(23)ui.1=∑β∈Js,niA2.ia˜.fββ∈ℂ1×n,f=1,…,n,(24)u.j2=∑α∈Is,njAA∗j.a˜l.αα∈ℂn×1,l=1,…,n,are the row and column vectors, respectively. Here, a˜.f and a˜l. are the fth column and lth row of A˜≔A2A∗.Proof.
Taking into account (20), we have for A○#(25)aij○#,r=∑l=1n∑f=1nail#alfafj†.
By substituting (15) and (12) in (25), we obtain(26)aij○#,r=∑l=1n∑f=1n∑β∈Js,niA2.ia.fββafl∑β∈Js,nA2ββ∑α∈Is,njAA∗j.al.∗αα∑α∈Is,nAA∗αα=∑f=1n∑l=1n∑β∈Js,njA2.je.fββa˜fl∑α∈Is,njAA∗j.el.αα∑β∈Js,nA2ββ∑α∈Is,nAA∗αα,where e.l and el. are the unit column and row vectors, respectively, such that all their components are 0, except the lth components which are 1; a˜lf is the lfth element of the matrix A˜≔A2A∗.
Let(27)uil1≔∑f=1n∑β∈Js,niA2.ie.fββa˜fl=∑β∈Js,niA2.iA˜.lββ,i,l=1,…,n.
Construct the matrixU1=uil1∈ℍn×n. It follows that(28)∑luil1∑α∈Is,njAA∗j.el.αα=∑α∈Is,njAA∗j.ui.1αα,where ui.1 is the ith row of U1. So, we get (22).
If we first consider(29)uif2≔∑la˜fl∑α∈Is,njAA∗j.el.αα=∑α∈Is,njAA∗j.a˜f.αα,f,j=1,…,n,and construct the matrix U2=uif2∈ℍn×n, then from(30)∑f=1n∑β∈Js,niA2.ie.fββuif2=∑β∈Js,niA2.iu.f2ββ,it follows (22).Taking into account (21), the following theorem on the determinantal representation of the left core inverse can be proved similarly.Theorem 3.
LetA∈ℂnCM and rkA2=rkA=s. Then, for its left core inverse matrix A○#=aij○#,l, we have(31)aij○#,l=∑α∈Is,njA2j.vi.1αα∑β∈Js,nA∗Aββ∑α∈Is,nA2αα=∑β∈Js,niA∗A.iv.j2ββ∑β∈Js,nA∗Aββ∑α∈Is,nA2αα,where(32)vi.1=∑β∈Js,niA∗A.ia¯.fββ∈ℍ1×n,f=1,…,n,v.j2=∑α∈Is,njA2j.a¯l.αα∈ℍn×1,l=1,…,n.
Here,a¯.f and a¯l. are the fth column and lth row of A¯≔A∗A2.


## 4. Determinantal Representations of the Core-EP Inverses

Similar to [4], we introduce two core-EP inverses.Definition 4.
A matrixX∈ℂn×n is said to be the right core-EP inverse of A∈ℂn×n if it satisfies the conditions(33)XAX=A,CX=CX∗=CAd.
It is denoted byA○†.Definition 5.
A matrixX∈ℂn×n is said to be the left core-EP inverse of A∈ℂn×n if it satisfies the conditions(34)XAX=A,ℛX=ℛX∗=ℛAd.
It is denoted byA○†.Remark 5.
SinceCA∗d=ℛAd, then the left core-EP inverse A○† of A∈ℂn×n is similar to the ∗core inverse introduced in [4], and the dual core-EP inverse introduced in [42].
According to [4], we have the following representations the core-EP inverses of A∈ℂn×n:(35)A○†=A2,3,6a,CA○†⊆CAk,A○†=A2,4,6b,ℛA○†⊆ℛAk.
Thanks to [42], the following representations of the core-EP inverses will be used for their determinantal representations.Lemma 3.
LetA∈ℂn×n and IndA=k. Then,(36)A○†=AkAk+1†,(37)A○†=Ak+1†Ak.
Moreover, ifIndA=1, then we have the following representations of the right and left core inverse matrices:(38)A○#=AA2†,(39)A○#=A2†A.Theorem 4.
SupposeA∈ℂn×n, IndA=k, and rkAk=s, and there exist A○† and A○†. Then, A○†=aij○†,r and A○†=aij○†,l possess the determinantal representations, respectively,(40)aij○†,r=∑α∈Is,njAk+1Ak+1∗j.a^i.αα∑α∈Is,nAk+1Ak+1∗αα,(41)aij○†,l=∑β∈Js,niAk+1∗Ak+1.iaˇ.jββ∑β∈Js,nAk+1∗Ak+1ββ,where a^i. is the ith row of A^=AkAk+1∗ and aˇ.j is the jth column of Aˇ=Ak+1∗Ak.Proof.
ConsiderAk+1†=aijk+1,† and Ak=aijk. By (36),(42)aij○†,r=∑t=1naitkatjk+1,†.
Taking into account (10) for the determinantal representation of Ak+1†, we get(43)aij○†,r=∑t=1naitk∑α∈Is,njAk+1Ak+1∗j.at.k+1,†αα∑α∈Ir,mAk+1Ak+1∗αα,where at.k+1 is the tth row of Ak+1∗. Since ∑t=1naitkat.k+1=a^i., then it follows (40).
The determinantal representation (41) can be obtained similarly by integrating (9) for the determinantal representation of Ak+1† in (37).Taking into account the representations (38)–(39), we derive the determinantal representations of the right and left core inverse matrices that have simpler expressions than those obtained in Theorems 2 and 3.Corollary 3.
LetA∈ℂsn×n and IndA=1, and there exist A○# and A○#. Then, A○#=aij○#,r and A○#=aij○#,l can be expressed as follows:(44)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 ith row of A^=AA2∗ and aˇ.j is the jth column of Aˇ=A2∗A.


## 5. Determinantal Representations of the DMP and MPD Inverses

The concept of the DMP inverse in complex matrices was introduced in [6] by Malik and Thome.Definition 6.
[6] Suppose A∈ℂn×n and IndA=k. A matrix X∈ℂn×n is said to be the DMP inverse of A if it satisfies the conditions(45)XAX=X,XA=AdA,AkX=AkA†.
It is denoted byAd,†.
According to [6], if an arbitrary matrix satisfies the system of equations (45), then it is unique and has the following representation:(46)Ad,†=AdAA†.Theorem 5.
LetA∈ℂsn×n, IndA=k, and rkAk=s1. Then, its DMP inverse Ad,†=aijd,† has the following determinantal representations:(47)aijd,†=∑α∈Is,njAA∗j.ui.1αα∑β∈Js1,nAk+1ββ∑α∈Is,nAA∗αα=∑β∈Js1,niAk+1.iu.j2ββ∑β∈Js1,nAk+1ββ∑β∈Js,nAA∗ββ,where(48)ui.1=∑β∈Js1,niAk+1.ia˜.fββ∈ℂ1×n,f=1,…,n,u.j2=∑α∈Is,njAA∗j.a˜.lαα∈ℂn×1,l=1,…,n.
Here,a˜.f and a˜l. are the fth column and the lth row of A˜≔Ak+1A∗.Proof.
Taking into account (46) for Ad,†, we get(49)aijd,†=∑l=1n∑f=1naildalfafj†.
By substituting (14) and (10) for the determinantal representations of Ad and A† in (49), we get(50)aijd,†=∑l=1n∑f=1n∑β∈Js1,niAk+1.ia.lkββ∑β∈Js1,nAk+1ββalf∑α∈Is,njAA∗j.af.∗αα∑α∈Is,nAA∗αα=∑l=1n∑f=1n∑s1,nAk+1.ie.lββ∑β∈Js1,nAk+1ββa˜lf∑α∈Is,njAA∗j.ef.αα∑α∈Is,nAA∗αα,where e.l and el. are the lth unit column and row vectors and a˜lf is the lfth element of the matrix A˜=Ak+1A∗. If we put(51)uif1≔∑l=1n∑β∈Js1,niAk+1.ie.lββa˜lf=∑β∈Js1,niAk+1.ia˜.fββ,as the fth component of the row vector ui.1=ui11,…,uin1. Then from(52)∑f=1nuif1∑α∈Is,njAA∗j.ef.αα=∑α∈Is,njAA∗j.ui.1αα,it follows (47). If we initially obtain(53)ulj2≔∑f=1na˜lf∑α∈Is,njAA∗j.ef.αα=∑α∈Is,njAA∗j.a˜l.αα,the lth component of the column vector u.j2=u1j2,…,unj2. Then from(54)∑l=1n∑β∈Js1,niAk+1.ie.lββulj2=∑β∈Js1,niAk+1.iu.j2ββ,it follows (47).The name of the DMP inverse is in accordance with the order of using the Drazin inverse (D) and the Moore-Penrose (MP) inverse. In that connection, it would be logical to consider the following definition.Definition 7.
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(55)XAX=X,AX=AAd,XAk=A†Ak.
It is denoted byA†,d.
Similar as for the DMP inverse, it can be proved that the matrixA†,d is unique, and it can be represented as(56)A†,d=A†AAd.Theorem 6.
LetA∈ℂsn×n, IndA=k, and rkAk=s1. Then, its MPD inverse A†,d=aij†,d has the following determinantal representations:(57)aij†,d=∑β∈Js,niA∗A.iv.j1ββ∑β∈Js,nA∗Aββ∑β∈Is1,nAk+1αα=∑α∈Is1,njAk+1j.vi.2αα∑α∈Is1,nA∗Aββ∑α∈Is,nAk+1αα,where(58)v.j1=∑α∈Is1,njAk+1j.a^l.αα∈ℂn×1,l=1,…,n,vi.2=∑β∈Js,niA∗A.ia^.fββ∈ℂ1×n,l=1,…,n.
Here,a^l. and a^.f are the lth row and the fth column of A^≔A∗Ak+1.Proof.
The proof is similar to the proof of Theorem 6.


## 6. Determinantal Representations of the CMP Inverse

Definition 8 [7].
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 4 [7].
LetA∈ℂn×n. The matrix X=Ac,† is the unique matrix that satisfies the following system of equations:(59)
X
A
X
=
X
,
A
X
A
=
A
1
,
A
X
=
A
1
A
†
,
X
A
=
A
†
A
1
.Moreover,(60)Ac,†=QAAdPA.Taking into account (60), it follows the next theorem about determinantal representations of the quaternion CMP inverse.Theorem 7.
LetA∈ℂsn×n, IndA=m, and rkAm=s1. Then, the determinantal representations of its CMP inverse Ac,†=aijc,† can be expressed as(61)aijc,†=∑β∈Js,niA∗A.iv.jlββ∑β∈Js,nA∗Aββ2∑β∈Js1,nAm+1ββ=∑α∈Is,njAA∗j.wi.lαα∑α∈Is,nAA∗αα2∑β∈Js1,nAm+1ββ,for all l=1,2, where(62)v.j1=∑α∈Is,njAA∗j.u^t.αα∈ℂn×1,t=1,…,n,(63)wi.1=∑β∈Js,niA∗A.iu^.kββ∈ℂ1×n,k=1,…,n,(64)v.j2=∑α∈Is,njA∗Aj.g˜t.αα∈ℂn×1,t=1,…,n,(65)wi.2=∑β∈Js,niA∗A.ig˜.kββ∈ℂ1×n,k=1,…,n.
Here,u^t. is the tth row and u^.k is the kth column of U^≔UAA∗, g˜t. is the tth row and g˜.k is the kth column of G˜≔A∗AG, and the matrices U=uij∈ℂn×n and G=gij∈ℂn×n are such that(66)uij=∑α∈Is1,njAm+1j.a^i.αα,gij=∑β∈Js1,niAm+1.ia˜.jββ,where a^i. is the ith row of A^≔A∗Am+1 and a˜.j is the jth column of A˜≔Am+1A∗.Proof.
SupposeQA=qilA, Ad=aild, and PA=pilA. Taking into account (60), we get(67)aijc,†=∑l=1n∑k=1nqilAalkdpkjA.(a)
Taking into account the expressions (14), (12), and (13) for the determinantal representations of Ad, QA, and PA, respectively, we have(68)aijc,†=∑l∑t∑β∈Js,niA∗A.ia˙.tββ∑β∈Js,nA∗Aββ∑α∈Is1,nlAm+1l.at.mαα∑α∈Is1,nAm+1αα×∑α∈Is,njAA∗j.a¨l.αα∑α∈Is,nAA∗αα,wherea˙.t is the tth column of A∗A, a¨l. is the lth row of AA∗, and at.m is the tth row of Am. So, it is clear that(69)aijc,†=∑l∑t∑k∑β∈Js,niA∗A.ie.tββa^tk∑α∈Is1,nlAm+1l.ek.αα∑β∈Js,nA∗Aββ∑α∈Is1,nAm+1αα×∑α∈Is,njAA∗j.a¨l.αα∑α∈Is,nAA∗αα,wheree.t is the tth unit column vector, ek. is the kth unit row vector, and a^tk is the tkth element of A^=A∗Am+1. Denote(70)utl≔∑ka^tk∑α∈Is1,njAm+1l.ek.αα=∑α∈Is1,njAm+1l.a^t.αα,thetth component of a column-vector u.l=u1l,…,unl. Since(71)∑t∑β∈Js,niA∗A.ie.tββutl=∑β∈Js,niA∗A.iu.lββ,then(72)aijc,†=∑l∑β∈Js,niA∗A.iu.lββ∑α∈Is,njAA∗j.a¨l.αα∑β∈Jr,nA∗Aββ∑α∈Is1,nAm+1αα∑α∈Is,nAA∗αα.Construct the matrixU=utl∈ℂn×n, where utl is given by (70), and denote U^≔UAA∗. Then, taking into account that A∗Aββ=AA∗αα, we have(73)aijc,†=∑t∑k∑β∈Js,niA∗A.ie.tββu^tk∑α∈Is,njAA∗j.ek.αα∑β∈Js,nA∗Aββ2∑α∈Is1,nAm+1αα.If we put that(74)vtj1≔∑ku^tk∑α∈Is,njAA∗j.ek.αα=∑α∈Is,njAA∗j.u^t.αα,is thetth component of a column vector, v.j1=v1j1,…,vnj1. Then from(75)∑t∑β∈Js,niA∗A.ie.tββvtj1=∑β∈Js,niA∗A.iv.j1ββ,it follows (61) with v.j1 given by (62). If we initially put(76)wik1≔∑t∑β∈Js,niA∗A.ie.tββu^tk=∑β∈Js,niA∗A.iu^.kββ,as thekth component of the row vector, wi.1=wi11,…,win1. Then from(77)∑kwik1∑α∈Is,njA2j.ek.αα=∑α∈Is,njA2j.wi.1αα,it follows (61) with wi.1 given by (64).(b)
By using the determinantal representation (14) for Ad in (67), we have(78)aijc,†=∑k∑t∑β∈Js,niA∗A.ia˙.tββ∑β∈Js,nA∗Aββ∑β∈Js1,ntAm+1.ta.kmββ∑β∈Js,nAm+1ββ×∑α∈Is,njAA∗j.a¨k.αα∑α∈Is,nAA∗αα.Therefore,(79)aijc,†=∑l∑k∑t∑β∈Js,niA∗A.ia˙.tββ∑β∈Js,nA∗Aββ×∑β∈Js1,ntAm+1.te.kββ∑β∈Js1,nAm+1ββa˜kl∑α∈Is,njAA∗j.el.αα∑α∈Is,nAA∗αα,wheree.k is the kth unit column vector, el. is the lth unit row vector, and a˜kl is the klth element of A˜=Am+1A∗.Denote by(80)gtl≔∑l∑β∈Js1,ntAm+1.te.kββa˜kl=∑β∈Js1,ntAm+1.ta˜.lββ,thelth component of a row-vector gt.=gt1,…,gtn. Then,(81)∑lgtl∑α∈Is,njAA∗j.el.αα=∑α∈Is,njAA∗j.gt.αα.From this, it follows that(82)aijc,†=∑t∑β∈Js,niA∗A.ia˙.tββ∑α∈Is,njAA∗j.gt.αα∑β∈Jr,nA∗Aββ∑α∈Is1,nAm+1αα∑α∈Is,nAA∗αα.Construct the matrixG=gtl∈ℂn×n, where gtl is given by (80). Denote G˜≔A∗AG. Then(83)aijc,†=∑t∑k∑β∈Js,niA∗A.ie.tββu^tk∑α∈Is,njAA∗j.ek.αα∑β∈Js,nA∗Aββ2∑α∈Is1,nAm+1αα.Similar to the above, if we denote by(84)vtj1≔∑ku^tk∑α∈Is,njAA∗j.ek.αα=∑α∈Is,njAA∗j.u^t.αα,thetth component of a column vector v.j2=v1j2,…,vnj2, then(85)∑t∑β∈Js,niA∗A.ie.tββvtj1=∑β∈Js,niA∗A.iv.j1ββ.Thus, we have (61) withv.j2 given by (64).If, now, we denote by(86)wik1≔∑t∑β∈Js,niA∗A.ie.tββu^tk=∑β∈Js,niA∗A.iu^.kββ,thekth component of the row vector wi.1=wi11,…,win1. Then, from(87)∑kwik1∑α∈Is,njA2j.ek.αα=∑α∈Is,njA2j.wi.1αα.
So, finally, we have (61) with wi.2 given by (65).Remark 6.
Concerning the possible existence of dual to the CMP inverse, we note the following. Taking into accountAd2A=Ad,(88)PAAdQA=AA†Ad2AA†A=AA†Ad=AA†AAd2=Ad.


## 7. Cramer’s Rule for Some Constrained Linear Equations

Cramer’s rules for special solutions to systems of linear equations stand as possible applications of determinantal representations of the core inverse and its generalizations. For instance, consider Cramer’s rule for the system which has the important applied significance.First, note that the core inverse has close relation with the Bott-Duffin inverse. According to [19], the Bott-Duffin inverse of A∈ℍn×n with respect to CA can be given by(89)ACA−1=PAA−InPA+In−1,where A−InPA+In needs to be nonsingular. According to [1], ACA−1 coincides with the right core inverse A○#. One can find more details regarding the Bott-Duffin inverse in [44].Consider the constrained linear equations(90)Ax+y=b,x∈CA,y∈NA,where A∈ℂn×n and b∈ℂn. According to [45], this equation arises in electrical networks and its solution is determined by the following lemma.Lemma 5.
LetA−InPA+In be nonsingular. Then, equation (90) has for every b the unique solution(91)
x
=
A
C
A
−
1
b
,
(92)
y
=
I
−
A
A
C
A
−
1
b
.The following theorem gives Cramer’s rule of finding the solution (91).Theorem 8.
Suppose thatACA−1 exists. Then,(93)xi=∑β∈Js,niA2.ib˜ββ∑β∈Js,nA2ββ∑α∈Is,nAA∗αα,where b˜≔U2b, U2=uij2 is such that(94)uij2=∑α∈Is,njAA∗j.a˜i.αα,and a˜i. are the ith row of A˜≔A2A∗.Proof.
Supposeb=b1,…,bnT and x=x1,…,xnT. Since the Bott-Duffin inverse ACA−1=aij−1 coincides with the right core inverse A○#, then its determinantal representation can be obtained by Theorem 2. We use the determinantal representation (22). So, for all i=1,…,n, we have(95)xi=∑kaik−1bk=∑kaik○#bk=∑k∑β∈Js,niA2.iu.k2ββbk∑β∈Js,nA2ββ∑α∈Is,nAA∗αα.
Construct the matrixU2 with columns determined by (24) and denote b˜≔U2b. Then, from using this denotation in (95), it follows (93).It is evident that the solutiony=y1,…,ynT from (92) by the components can be expressed as(96)yi=bi−x˜i,where x˜i is the ith component of the column vector x˜=Ax.


## 8. An Example

Consider the matrix(97)A=200−iii−i−i−i.Since(98)AA∗=42i2i−2i3−1−2i−13,A2=4002−2i00−2−2i00,A3=8004−4i00−4−4i00,then rkA=2 and rkA2=rkA3=1, then k=IndA=2 and r1=1. So, we shall find A○† and A○† by (40) and (41), respectively. Since(99)A^=A2A3∗=1621+i−1+i1−i1i−1−ii1,then by (40)(100)a11○†,r=∑α∈I1,31A3A3∗1.a^1.αα∑α∈I1,3A3A3∗αα=14.By similarly continuing, we get(101)A○†=1821+i−1+i1−i1i−1−ii1.By analogy, according to (41), we have(102)A○†=12100000000.The DMP inverseAd,† can be find by Theorem 5. Since(103)A˜=A3A∗=442i2i2−2i1+i1+i−2−2i1−i1−i,and rkA3=1, then(104)u11=a˜1.,u21=a˜2.,u31=a˜3..Furthermore, by (47),(105)a11d,†=∑α∈I2,31AA∗1.u1.1αα∑β∈J1,3A3ββ∑α∈I2,3AA∗αα=1192det168i−2i3+det168i−2i3=13.By similarly continuing, we get(106)Ad,†=11242i2i2−2i1+i1+i−2−2i1−i1−i.Similarly by Theorem 6, we get(107)A†,d=14200−i00−i00.Finally, by Theorem, we find the CMP inverseAc,†=aijc,†. Since rkA3=1, then G=A˜ and(108)G˜=A∗AA˜=1663i3i−2i11−2i11.Furthermore, by (65),(109)w112=∑β∈J2,31A∗A.1g˜.1ββ=det60−2i2+det60−2i2=24.By similar calculations, we get(110)w1.2=384,96i,96i,w2.2=−192i,96,96,w3.2=−192i,96,06.So, by (61), we get(111)a11c,†=∑α∈I2,31AA∗1.w1.2αα∑α∈I2,3AA∗αα2∑β∈J1,3A3ββ=14608det384192i−2i3+det384192i−2i3=13.By similarly continuing, we derive(112)Ac,†=11242i2i−2i11−2i11.


## 9. Conclusions

In this chapter, we get the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived. Determinantal representation the Bott-Duffin inverse and its application to get Cramer’s rule of the solution to the constrained linear equations are obtained.



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*Source: 1631979-2019-10-01.xml*