1. Introduction

In the whole chapter, the notations R 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 a rank r. For A∈Cm×n, the symbols A∗ and rkA specify 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:

AXA=AE1

XAX=XE2

AX∗=AXE3

XA∗=XAE4

For A∈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 Eq. (2) and the following equations,

AX=XA;E5

XAk+1=AkE6

Ak+1X=Ak.E7

In particular, if IndA=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 (6) and (7) 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 matrix A satisfying the conditions i,j,… is called an ij…-inverse of A, and is denoted by Aij…. The set of matrices Aij… is denoted Aij…. In particular, A1 is called the inner inverse, A2 is called the outer inverse, A12 is called the reflexive inverse, A1,2,3,4 is the Moore-Penrose inverse, etc.

For an arbitrary matrix A∈Cm×n, we denote by

NA=x∈Hn×1:Ax=0, the kernel (or the null space) of A;
CA=y∈Hm×1:y=Axx∈Hn×1, the column space (or the range space) of A; and
RA=y∈H1×n:y=xAx∈H1×m, the row space of A.

PA≔AA† 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 S. Malik in [2] and S.Z. Xu et al. in [3], among others.

Definition 1.1. [1] A matrix X∈Cn×n is called the core inverse of A∈Cn×n if it satisfies the conditions

AX=PA,andCX=CA.

When such matrix X exists, it is denoted as A○#.

In 2014, the core inverse was extended to the core-EP inverse defined by K. Manjunatha Prasad and K.S. Mohana [4]. Other generalizations of the core inverse were recently introduced for n×n complex matrices, namely BT inverses [5], DMP inverses [2], CMP inverses [6], etc. The characterizations, computing methods, and some applications of the core inverse and its generalizations were recently investigated in complex matrices and rings (see, e.g., [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 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, 20, 21, 22, 23, 24, 25]). In this chapter, we get new determinantal representations of the core inverse and its generalizations using recently obtained by the author determinantal representations of the Moore-Penrose inverse and the Drazin inverse over the quaternion skew field, and over the field of complex numbers as a special case [26, 27, 28, 29, 30, 31, 32, 33, 34]. 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 an reflexive inverse obtained in [19, 20].

The chapter is organized as follows: in Section 2, we start with preliminary introduction of determinantal representations of the Moore-Penrose inverse and the Drazin inverse. In Section 3, we give determinantal representations of the core inverse and its generalizations, namely the right and left core inverses are established in Section 3.1, the core-EP inverses in Section 3.2, the core DMP inverse and its dual in Section 3.3, and finally the CMP inverse in Section 3.4. A numerical example to illustrate the main results is considered in Section 4. Finally, in Section 5, the conclusions are drawn.

2. Preliminaries

Let α≔α1…αk⊆1…m and β≔β1…βk⊆1…n be subsets with 1≤k≤minmn. By Aβα, we denote a submatrix of A∈Hm×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

Lk,n≔α:α=α1…αk1≤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,mi∈α and Jr,nj≔β:β∈Lr,nj∈β.

The jth columns and the ith rows of A and A∗ denote a.j and a.j∗ and ai. and ai.∗, 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 2.1. [28] IfA∈Hrm×n, then the Moore-Penrose inverseA†=aij†∈Cn×mpossesses the determinantal representations

aij†=∑β∈Jr,niA∗A.ia.j∗ββ∑β∈Jr,nA∗Aββ=E8

=∑α∈Ir,mjAA∗j.ai.∗αα∑α∈Ir,mAA∗αα.E9

Remark 2.2. For an arbitrary full-rank matrix A∈Crm×n, a row vector b∈H1×m, and a column-vector c∈Hn×1, we put, respectively,

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 2.3. [21] LetA∈Crm×n. Then, the following determinantal representations can be obtained

for the projector QA=qijn×n,
qij=∑β∈Jr,niA∗A.iȧ.jββ∑β∈Jr,nA∗Aββ=∑α∈Ir,njA∗Aj.ȧi.αα∑α∈Ir,nA∗Aαα,E10
where ȧ.j is the jth column and ȧi. is the ith row of A∗A; and
for the projector PA=pijm×m,
pij=∑α∈Ir,mjAA∗j.a¨i.αα∑α∈Ir,mAA∗αα=∑β∈Jr,miAA∗.ia¨.jββ∑β∈Jr,mAA∗ββ,E11
where a¨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 2.4. [21] LetA∈Cn×nwithIndA=kandrkAk+1=rkAk=r. Then, the determinantal representations of the Drazin inverseAd=aijd∈Cn×nare

aijd=∑β∈Jr,niAk+1.ia.jkββ∑β∈Jr,nAk+1ββ=E12

=∑α∈Ir,njAk+1j.ai.kαα∑α∈Ir,nAk+1αα,E13

where ai.k is the ith row and a.jk is the jth column of Ak.

Corollary 2.5. [21] LetA∈Cn×nwithIndA=1andrkA2=rkA=r. Then, the determinantal representations of the group inverseA#=aij#∈Cn×nare

aij#=∑β∈Jr,niA2.ia.jββ∑β∈Jr,nA2ββ=∑α∈Ir,njA2j.ai.αα∑α∈Ir,nA2αα.E14

3. Determinantal representations of the core inverse and its generalizations

3.1 Determinantal representations of the core inverses

Together with the core inverse in [35], the dual core inverse was to be introduced. Since the both these core inverses are equipollent and they are different only in the position relative to the inducting matrix A, we propose called them as the right and left core inverses regarding to their positions. So, from [1], we have the following definition that is equivalent to Definition 1.1.

Definition 3.1. A matrix X∈Cn×n is said to be the right core inverse of A∈Cn×n if it satisfies the conditions

AX=PA,andCX=CA.

When such matrix X exists, it is denoted as A○#.

The following definition of the left core inverse can be given that is equivalent to the introduced dual core inverse [35].

Definition 3.2 A matrix X∈Cn×n is said to be the left core inverse of A∈Cn×n if it satisfies the conditions

XA=QA,andRX=RA.E15

When such matrix X exists, it is denoted as A○#.

Remark 3.3. In [35], the conditions of the dual core inverse are given as follows:

A○#A=PA∗,andCA○#⊆CA∗.

Since PA∗=A∗A∗†=A†A∗=A†A=QA, and RA=CA∗, then these conditions and (15) are analogous.

Due to [1], we introduce the following sets of quaternion matrices

CnCM=A∈Cn×n:rkA2=rkA,CnEP=A∈Cn×n:A†A=AA†=CA=CA∗.

The matrices from CnCM 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. Due to [1], we have the following representations of the right and left core inverses.

Lemma 3.4. [1] LetA∈CnCM. Then,

A○#=A#AA†,E16

A○#=A†AA#E17

Remark 3.5. In Theorems 3.6 and 3.7, we will suppose that A∈CnCM but A∉CnEP. Because, if A∈CnCM and A∈CnEP (in particular, A is Hermitian), then from Lemma 3.4 and the definitions of the Moore-Penrose and group inverses, it follows that A○#=A○#=A#=A†.

Theorem 3.6.LetA∈CnCMandrkA2=rkA=s. Then, its right core inverse has the following determinantal representations

aij○#,r=∑α∈Is,njAA∗j.ui.1αα∑β∈Js,nA2ββ∑α∈Is,nAA∗αα=E18

=∑β∈Js,niA2.iu.j2ββ∑β∈Js,nA2ββ∑α∈Is,nAA∗αα,E19

where

ui.1=∑β∈Js,niA2.ia˜.fββ∈C1×n,f=1,…,nu.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 (16), we have for #A,

aij#,r=∑l=1n∑f=1nail#alfafj†.E20

By substituting (14) and (15) in (20), we obtain

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 (lf)th element of the matrix A˜≔A2A∗.

Let

uil1≔∑f=1n∑β∈Js,niA2.ie.fββa˜fl=∑β∈Js,niA2.ia˜.lββ,i,l=1,…,n.

Construct the matrix U1=uil1∈Hn×n. It follows that

∑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 (18). If we first consider

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

∑f=1n∑β∈Js,niA2.ie.fββuif2=∑β∈Js,niA2.iu.f2ββ,

it follows (19).□

Taking into account (17), the following theorem on the determinantal representation of the left core inverse can be proved similarly.

Theorem 3.7.LetA∈CnCMandrkA2=rkA=s. Then for its left core inverse#A=aij#,l, we have

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

vi.1=∑β∈Js,niA∗A.ia¯.fββ∈C1×n,f=1,…,n

v.j2=∑α∈Is,njA2j.a¯l.αα∈Cn×1,l=1,…,n.

Here a¯.f and a¯l. are the fth column and lth row of A¯≔A∗A2.

3.2 Determinantal representations of the core-EP inverses

Similar as in [4], we introduce two core-EP inverses.

Definition 3.8. A matrix X∈Cn×n is said to be the right core-EP inverse of A∈Cn×n if it satisfies the conditions

XAX=A,andCX=CX∗=CAd.

It is denoted as A○†.

Definition 3.9. A matrix X∈Cn×n is said to be the left core-EP inverse of A∈Cn×n if it satisfies the conditions

XAX=A,andRX=RX∗=RAd.

It is denoted as A○†.

Remark 3.10. Since CA∗d=RAd, then the left core inverse A○† of A∈Cn×n is similar to the ∗core inverse introduced in [4], and the dual core-EP inverse introduced in [35].

Due to [4], we have the following representations the core-EP inverses of A∈Cn×n,

A○†=A2,3,6aandCA○†⊆CAk,A○†=A2,4,6bandRA○†⊆RAk.

Thanks to [35], the following representations of the core-EP inverses will be used for their determinantal representations.

Lemma 3.11.LetA∈Cn×nandIndA=k. Then

A○†=AkAk+1†,E21

A○†=Ak+1†Ak.E22

Moreover, if IndA=1, then we have the following representations of the right and left core inverses

A○#=AA2†,E23

A○#=A2†A.E24

Theorem 3.12.SupposeA∈Cn×n,IndA=k,rkAk=s, and there existA○†andA○†. ThenA○†=aij○†,randA○†=aij○†,lpossess the determinantal representations, respectively,

aij○†,r=∑α∈Is,njAk+1Ak+1∗j.âi.αα∑α∈Is,nAk+1Ak+1∗αα,E25

aij○†,l=∑β∈Js,niAk+1∗Ak+1.iaˇ.jββ∑β∈Js,nAk+1∗Ak+1ββ,E26

where âi. is the ith row of Â=AkAk+1∗ and aˇ.j is the jth column of Aˇ=Ak+1∗Ak.

Proof. Consider Ak+1†=aijk+1† and Ak=aijk. By (21),

aij○†,r=∑t=1naitkatjk+1†.

Taking into account (9) for the determinantal representation of Ak+1†, we get

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∗=âi., then it follows (25).

The determinantal representation (26) can be obtained similarly by integrating (8) for the determinantal representation of Ak+1† in (22).□

Taking into account the representations (23)-(24), we obtain the determinantal representations of the right and left core inverses that have more simpler expressions than they are obtained in Theorems 3.6 and 3.7.

Corollary 3.13.LetA∈Csn×n,IndA=1, and there existA○#andA○#. ThenA○#=aij○#,randA○#=aij○#,lcan be expressed as follows

aij○#,r=∑α∈Is,njA2A2∗j.âi.αα∑α∈Is,nA2A2∗αα,aij○#,l=∑β∈Js,niA2∗A2.iaˇ.jββ∑β∈Js,nA2∗A2ββ,

where âi. is the ith row of Â=AA2∗ and aˇ.j is the jth column of Aˇ=A2∗A.

3.3 Determinantal representations of the DMP and MPD inverses

The concept of the DMP inverse in complex matrices was introduced in [2] by S. Malik and N. Thome.

Definition 3.14. [2] 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

XAX=X,XA=AdA,andAkX=AkA†.E27

It is denoted as Ad,†.

Due to [2], if an arbitrary matrix satisfies the system of Eq. (27), then it is unique and has the following representation

Ad,†=AdAA†.E28

Theorem 3.15.LetA∈Csn×n,IndA=k, andrkAk=s1. Then, its DMP inverseAd,†=aijd,†has the following determinantal representations.

aijd,†=∑α∈Is,njAA∗j.ui.1αα∑β∈Js1,nAk+1ββ∑α∈Is,nAA∗αα=E29

=∑β∈Js1,niAk+1.iu.j2ββ∑β∈Js1,nAk+1ββ∑β∈Js,nAA∗ββ,E30

where

ui.1=∑β∈Js1,niAk+1.ia˜.fββ∈C1×n,f=1,…,n,u.j2=∑α∈Is,njAA∗j.a˜l.αα∈Cn×1,l=1,…,n.

Here, a˜.f and âl. are the fth column and the lth row of A˜≔Ak+1A∗.

Proof. Taking into account (28) for Ad,†, we get

aijd,†=∑l=1n∑f=1naildalfafj†.E31

By substituting (12) and (9) for the determinantal representations of Ad and A† in (31), we get

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∑β∈Js1,niAk+1.ie.lββ∑β∈Js1,nAk+1ββa˜lf∑α∈Is,nj(AA∗j.ef.αα∑α∈Is,nAA∗αα,E32

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

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

∑f=1nuif1∑α∈Is,njAA∗j.ef.αα=∑α∈Is,njAA∗j.ui.1αα,

it follows (29). If we initially obtain

ulj2≔∑f=1na˜lf∑α∈Is,njAA∗j.ef.αα=∑α∈Is,njAA∗j.a˜l.αα,

as the lth component of the column vector u.j2=u1j2…unj2, then from

∑l=1n∑β∈Js1,niAk+1.ie.lββulj2=∑β∈Js1,niAk+1.iu.j2ββ,

it follows (30).□

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 3.16. Suppose A∈Cn×n and IndA=k. A matrix X∈Cn×n is said to be the MPD inverse of A if it satisfies the conditions

XAX=X,AX=AAd,andXAk=A†Ak.

It is denoted as A†,d.

The matrix A†,d is unique, and it can be represented as

A†,d=A†AAd.E33

Theorem 3.17.LetA∈Csn×n,IndA=k, andrkAk=s1. Then, its MPD inverseA†,d=aij†,dhas the following determinantal representations

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

v.j1=∑α∈Is1,njAk+1j.âl.αα∈Cn×1,l=1,…,nvi.2=∑β∈Js,niA∗A.iâ.fββ∈C1×n,l=1,…,n.

Here, âl. and â.f are the lth row and the fth column of Â≔A∗Ak+1.

Proof. The proof is similar to the proof of Theorem 3.15.□

3.4 Determinantal representations of the CMP inverse

Definition 3.18. [6] Suppose A∈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 3.19. [6] LetA∈Cn×n. The matrixX=Ac,†is the unique matrix that satisfies the following system of equations:

XAX=X,AXA=A1,AX=A1A†,andXA=A†A1.

Moreover,

Ac,†=QAAdPA.E34

Taking into account (34), it follows the next theorem about determinantal representations of the quaternion CMP inverse.

Theorem 3.20.LetA∈Csn×n,IndA=m, andrkAm=s1. Then, the determinantal representations of its CMP inverseAc,†=aijc,†can be expressed as

aijc,†=∑β∈Js,niA∗A.iv.jlββ∑β∈Js,nA∗Aββ2∑β∈Js1,nAm+1ββE35

aijc,†=∑α∈Is,njAA∗j.wi.lαα∑α∈Is,nAA∗αα2∑β∈Js1,nAm+1ββE36

for all l=1,2, where

v.j1=∑α∈Is,njAA∗j.ût.αα∈Cn×1,t=1,…,n,E37

wi.1=∑β∈Js,niA∗A.iû.kββ∈C1×n,k=1,…,n,E38

v.j2=∑α∈Is,njA∗Aj.g˜t.αα∈Cn×1,t=1,…,n,E39

wi.2=∑β∈Js,niA∗A.ig˜.kββ∈C1×n,k=1,…,n.E40

Here, ût. is the tth row and û.k is the kth column of Û≔UAA∗, g˜t. is the tth row and g˜.k is the kth column of G˜≔A∗AG, and the matrices U=uij∈Hn×n and G=gij∈Hn×n are such that

uij=∑α∈Is1,njAm+1j.âi.αα,gij=∑β∈Js1,niAm+1.ia˜.jββ,

where âi. is the ith row of Â≔A∗Am+1 and a˜.j is the jth column of A˜≔Am+1A∗.

Proof. Taking into account (34), we get

aijc,†=∑l=1n∑k=1nqilAalkdpkjA,E41

where QA=qilA, Ad=aild, and PA=pilA.

a. Taking into account the expressions (13), (10), and (11) for the determinantal representations of Ad, QA, and PA, respectively, we have

aijc,†=∑l∑t∑β∈Js,niA∗A.iȧ.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,

where ȧ.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

aijc,†=∑l∑t∑k∑β∈Js,niA∗A.ie.tββâtk∑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,

where e.t is the tth unit column vector, ek. is the kth row vector, and âtk is the tkth element of Â=A∗Am+1.

Denote

utl≔∑kâtk∑α∈Is1,njAm+1l.ek.αα=∑α∈Is1,njAm+1l.ât.ααE42

as the tth component of a column vector u.l=u1l…unl. Then from

∑t∑β∈Js,niA∗A.ie.tββutl=∑β∈Js,niA∗A.iu.lββ,

we have

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 matrix U=utl∈Hn×n, where utl is given by (42), and denote Û≔UAA∗. Then, taking into account that A∗Aββ=AA∗αα, we have

aijc,†=∑t∑k∑β∈Js,niA∗A.ie.tββûtk∑α∈Is,njAA∗j.ek.αα∑β∈Js,nA∗Aββ2∑α∈Is1,nAm+1αα.

If we put that

vtj1≔∑kûtk∑α∈Is,njAA∗j.ek.αα=∑α∈Is,njAA∗j.ût.αα

is the tth component of a column vector v.j1=v1j1…vnj1, then from

∑t∑β∈Js,niA∗A.ie.tββvtj1=∑β∈Js,niA∗A.iv.j1ββ,

it follows (35) with v.j1 given by (37). If we initially put

wik1≔∑t∑β∈Js,niA∗A.ie.tββûtk=∑β∈Js,niA∗A.iû.kββ

as the kth component of the row vector wi.1=wi11…win1, then from

∑kwik1∑α∈Is,njA2j.ek.αα=∑α∈Is,njA2j.wi.1αα,

it follows (36) with wi.1 given by (38).

b. By using the determinantal representation (12) for Ad in (41), we have

aijc,†=∑k∑t∑β∈Js,niA∗A.iȧ.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,

aijc,†=∑l∑k∑t∑β∈Js,niA∗A.iȧ.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.

where e.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∗.

If we denote

gtl≔∑l∑β∈Js1,ntAm+1.te.kββa˜kl=∑β∈Js1,ntAm+1.ta˜.lββE43

as the lth component of a row vector gt.=gt1…gtn, then

∑lgtl∑α∈Is,njAA∗j.el.αα=∑α∈Is,njAA∗j.gt.αα.

From this, it follows that

aijc,†=∑t∑β∈Js,niA∗A.iȧ.tββ∑α∈Is,nj(AA∗j.gt.)αα∑β∈Jr,nA∗Aββ∑α∈Is1,nAm+1αα∑α∈Is,nAA∗αα.

Construct the matrix G=gtl∈Hn×n, where gtl is given by (43). Denote G˜≔A∗AG. Then,

aijc,†=∑t∑k∑β∈Js,niA∗A.ie.tββg˜tk∑α∈Is,njAA∗j.ek.αα∑β∈Js,nA∗Aββ2∑α∈Is1,nAm+1αα.

If we denote

vtj2≔∑kg˜tk∑α∈Is,njAA∗j.ek.αα=∑α∈Is,njAA∗j.g˜t.αα

as the tth component of a column vector v.j2=v1j2…vnj2, then

∑t∑β∈Js,niA∗A.ie.tββvtj2=∑β∈Js,niA∗A.iv.j2ββ.

Thus, we have (35) with v.j2 given by (39).

If, now, we denote

wik2≔∑t∑β∈Js,niA∗A.ie.tββg˜tk=∑β∈Js,niA∗A.ig˜.kββ

as the kth component of a row vector wi.2=wi12…win2, then

∑kwik2∑α∈Is,njAA∗j.ek.αα=∑α∈Is,njAA∗j.wi.2αα.

So, finally, we have (36) with wi.2 given by (40).

4. An example

Given the matrix

A=200−iii−i−i−i.

Since

AA∗=42i2i−2i3−1−2i−13,A2=4002−2i00−2−2i00,A3=8004−4i00−4−4i00,

then rkA=2 and rkA2=rkA3=1, and k=IndA=2 and r1=1. So, we shall find A○† and A○† by (25) and (26), respectively.

Since

Â=A2A3∗=1621+i−1+i1−i1i−1−ii1,

then by (25),

a11○†,r=∑α∈I1,31A3A3∗1.â1.αα∑α∈I1,3A3A3∗αα=14.

By similarly continuing, we get

A○†=1821+i−1+i1−i1i−1−ii1.

By analogy, due to (26), we have

A○†=12100000000.

The DMP inverse Ad,† can be found by Theorem 3.15. Since

A˜=A3A∗=442i2i2−2i1+i1+i−2−2i1−i1−i.

and rkA3=1, then

u11=a˜1.,u21=a˜2.,u31=a˜3..

Furthermore, by (29),

a11d,†=∑α∈I2,31AA∗1.u1.1αα∑β∈J1,3A3ββ∑α∈I2,3AA∗αα=1192det168i−2i3+det168i−2i3=13.

By similarly continuing, we get

Ad,†=11242i2i2−2i1+i1+i−2−2i1−i1−i.

Similarly by Theorem 3.17, we get

A†,d=14200−i00−i00.

Finally, by theorem, we find the CMP inverse Ac,†=aijc,†. Since rkA3=1, then G=A˜ and

G˜=A∗AA˜=1663i3i−2i11−2i11.

Furthermore, by (40),

w112=∑β∈J2,31A∗A.1g˜.1ββ=det60−2i2+det60−2i2=24.

By similar calculations, we get

w1.2=38496i96i,w2.2=−192i9696,w3.2=−192i9606.

So, by (36), we get

a11c,†=∑α∈I2,31AA∗1.w1.2αα∑α∈I2,3AA∗αα2∑β∈J1,3A3ββ=14608det384192i−2i3+det384192i−2i3=13.

By similarly continuing, we derive

Ac,†=11242i2i−2i11−2i11.

5. Conclusions

In this chapter, we get the direct method to find the core inverse and its generalizations that are based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, the DMP, MPD, and CMP inverses are derived.

Daun Jeong

Chapters authored

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