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# Quaternion MPCEP, CEPMP, and MPCEPMP Generalized Inverses

Written By

Ivan I. Kyrchei

Reviewed: 07 February 2022 Published: 23 April 2022

DOI: 10.5772/intechopen.103087

From the Edited Volume

## Matrix Theory - Classics and Advances

Edited by Mykhaylo Andriychuk

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## Abstract

A generalized inverse of a matrix is an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses exist for an arbitrary matrix and coincide with a regular inverse for invertible matrices. The most famous generalized inverses are the Moore–Penrose inverse and the Drazin inverse. Recently, new generalized inverses were introduced, namely the core inverse and its generalizations. Among them, there are compositions of the Moore–Penrose and core inverses, MPCEP (or MP–Core–EP) and EPCMP (or EP–Core–MP) inverses. In this chapter, the notions of the MPCEP inverse and CEPMP inverse are expanded to quaternion matrices and introduced new generalized inverses, the right and left MPCEPMP inverses. Direct method of their calculations, that is, their determinantal representations are obtained within the framework of theory of quaternion row-column determinants previously developed by the author. In consequence, these determinantal representations are derived in the case of complex matrices.

### Keywords

• Moore–Penrose inverse
• Drazin inverse
• generalized inverse
• core-EP inverse
• quaternion matrix
• noncommutative determinant

## 1. Introduction

The field of complex (or real) numbers is designated by C (R). The set of all m×n matrices over the quaternion skew field

H=h0+h1i+h2j+h3ki2=j2=k2=ijk=1h0h1h2h3R,

is represented by Hm×n, while Hrm×n is reserved for the subset of Hm×n with matrices of rank r. If h=h0+h1i+h2j+h3kH, its conjugate is h¯=h0h1ih2jh3k, and its norm h=hh¯=h¯h=h02+h12+h22+h32. For AHm×n, its rank and conjugate transpose are given by rankA and A, respectively. A matrix AHn×n is said to be Hermitian if A=A. Also,

• CrA=cHm×1:c=AddHn×1 is the right column space of A;

• RlA=cH1×n:c=dAdH1×m is the left row space of A;

• NrA=dHn×1:Ad=0 is the right null space of A;

• NlA=dH1×m:dA=0 is the left null space of A.

Let us recall the definitions of some well-known generalized inverses that can be extend to quaternion matrices as follows.

Definition 1.1. The Moore–Penrose inverse of AHn×m is the unique matrix A=X determined by equations

1AXA=A;2XAX=X;3AX=AX;4XA=XA.E1

Definition 1.2. The Drazin inverse of AHn×n is the unique Ad=X that satisfying Eq.(2) from (1) and the following equations,

5Ak=XAk+1,6XA=AX,

where k=IndA is the index of A, i.e. the smallest positive number such that rankAk+1=rankAk. If IndA1, then Ad=A# is the group inverse of A. If IndA=0, then A#=A=A1.

A matrix A satisfying the conditions i,j, is called an ij-inverse of A, and is denoted by Aij. In particular, A1 is called the inner inverse, A2 is called the outer inverse, and A12 is called the reflexive inverse, and A1,2,3,4 is the Moore–Penrose inverse, etc.

Note that the Moore–Penrose inverse inducts the orthogonal projectors PA=AA and QA=AA onto the right column spaces of A and A, respectively.

In [1], the core-EP inverse over the quaternion skew field was presented similarly as in [2].

Definition 1.3. The core-EP inverse of AHn×n is the unique matrix A=X which satisfies

According to [3], (Theorem 2.3), for mIndA, we have that A=AdAmAm. In a special case that IndA1, A=A# is the core inverse of A [4].

Definition 1.4. The dual core-EP inverse of AHn×n is the unique matrix A=X for which

Since the quaternion core-EP inverse A is related to the right space CrA of AHn×n and the quaternion dual core-EP inverse A is related to its left space RlA. So, in [1], they are also named the right and left core-EP inverses, respectively.

Various representations of core-EP inverse can be found in [1, 5, 6, 7]. In [8], continuity of core-EP inverse was investigated. Bordering and iterative methods to find the core-EP inverse were proved in [9, 10], and its determinantal representation for complex matrices was derived in [2]. New determinantal representations of the complex core-EP inverse and its various generalizations were obtained in [11]. The core-EP inverse was generalized to rectangular matrices [12], Hilbert space operators [13], Banach algebra elements [14], tensors [15], and elements of rings [3]. Combining the core-EP inverse or the dual core-EP inverse with the Moore–Penrose inverse, the MPCEP inverse and CEPMP inverse were introduced in [16] for bounded linear Hilbert space operators.

In the last years, interest in quaternion matrix equations is growing significantly based on the increasing their applications in various fields, among them, robotic manipulation [17], fluid mechanics [18, 19], quantum mechanics [20, 21, 22], signal processing [23, 24], color image processing [25, 26, 27], and so on.

The main goals of this chapter are investigations of the MPCEP and CEPMP inverses, introductions and representations of new right and left MPCEPMP inverses over the quaternion skew field, and obtaining of their determinantal representations as a direct method of their constructions. The chapter develops and continues the topic raised in a number of other works [28, 29, 30, 31, 32, 33], where determinantal representations of various generalized inverses were obtained.

The remainder of our chapter is directed as follows. In Section 2, we introduce of the quaternion MPCEP and CEPMP inverses and give characterizations of new generalized inverses, namely left and right MPCEPMP-inverses. In Section 3, we commence with introducing determinantal representations of the projection matrices inducted by the Moore–Penrose inverse and of core-EP inverse previously obtained within the framework of theory of quaternion row-column determinants and, based of them, determinantal representations of the MPCEP, CEPMP, and left and right MPCEPMP inverses are derived. Finally, the conclusion is drawn in Section 4.

## 2. Characterizations of the quaternion MPCEP, CEPMP, and MPCEPMP inverses

Analogously as in [16], the MPCEP inverse and CEPMP inverse can be defined for quaternion matrices.

Definition 2.1. Let AHn×n. The MPCEP (or MP-Core-EP) inverse of A is the unique solution A,=X to the system

X=XAX,XA=AAAA,AX=AA.

The CEPMP (or Core-EP-MP) inverse of A is the unique solution A,=X to the system

X=XAX,AX=AAAA,XA=AA.

We can represent the MPCEP inverse and CEPMP inverse, by [16], as

A,=AAA,E2
A,=AAA.E3

According to our concepts, we can define the left and right MPCPMP inverses.

Definition 2.2. Suppose AHn×n. The right MPCEPMP inverse of A is defined as

A,,,r=AAAAA.

The left MPCEPMP inverse of A is defined as

A,,,l=AAAAA.

The following gives the characteristic equations of these generalized inverses.

Theorem 2.3. Let A,XHn×n. The following statements are equivalent:

1. X is the right MPCEPMP inverse of A.

2. X=A,PA.E4

3. X is the unique solution to the three equations:

1.X=XAX,2.XA=A,A,3.AX=AAPA.E5

Proof. iii. By Eq. (2) and the denotation of PA, it is evident that

A,,,r=AAAAA=A,PA.

iiii. Now, we verify the condition (5). Let X=A,,,r=AAAAA. Then, from the Definition 1.1 and the representation (2), we have

XAX=AAAAAAAAAAA=AAAAAAAAA==AAAAAAA=AAAAA=X,XA=AAAAAA=A,A,AX=AAAAAA=AAPA.

To prove that the system (5) has unique solution, suppose that X and X1 are two solutions of this system. Then XA=A,A=X1A and AX=AAPA=AX1, which give XAX=XAX1=X1AX1=X1. Therefore, X is the unique solution to the system. □

The next theorem can be proved in the same way.

Theorem 2.4. Let A,XHn×n. The following statements are equivalent:

1. X is the left MPCEPMP inverse of A.

2. X=QAA,.E6

3. X is the unique solution to the system:

1.X=XAX,2.AX=AA,,3.XA=AA.

## 3. Determinantal representations of the quaternion MPCEP and *CEPMP inverses

It is well known that the determinantal representation of the regular inverse is given by the cofactor matrix. The construction of determinantal representations of generalized inverses is not so evident and unambiguous even for matrices with complex or real entries. Taking into account the noncommutativity of quaternions, this task is more complicated due to a problem of defining the determinant of a matrix with noncommutative elements (see survey articles [34, 35, 36] for detail). Only now, the solving this problem begins to be decided thanks to the theory of noncommutative column-row determinants introduced in [37, 38].

For arbitrary quaternion matrix AHn×n, there exists an exact technique to generate n row determinants (-determinants) and n column determinants (-determinants) by stating a certain order of factors in each term.

Definition 3.1. Let A=aijHn×n.

• For an arbitrary row index iIn, the ith -determinant of A is defined as

rdetiAσSn1nraiik1aik1ik1+1aik1+l1iaikrikr+1aikr+lrikr,

in which Sn denotes the symmetric group on In=1n, while the permutation σ is defined as a product of mutually disjunct subsets ordered from the left to right by the rules

σ=iik1ik1+1ik1+l1ik2ik2+1ik2+l2ikrikr+1ikr+lr,ikt<ikt+s,ik2<ik3<<ikr,t=2,,r,s=1,,lt.

• For an arbitrary column index jIn, the jth -determinant of A is defined as the sum

cdetjA=τSn1nrajkrjkr+lrajkr+1jkrajjk1+l1ajk1+1jk1ajk1j,

in which a permutation τ is ordered from the right to left in the following way:

τ=jkr+lrjkr+1jkrjk2+l2jk2+1jk2jk1+l1jk1+1jk1j,jkt<jkt+s,jk2<jk3<<jkr.

It is known that all - and -determinants are different in general. However, in [37], the following equalities are verified for a Hermitian matrix A that introduce a determinant of a Hermitian matrix: rdet1A==rdetnA=cdet1A==cdetnAdetAR.

D-Representations of various generalized inverses were developed by means of the theory of - and -determinants (see e.g. [28, 29, 30, 31]).

The following notations are used for determinantal representations of generalized inverses.

Let αα1αk1m and ββ1βk1n be subsets with 1kminmn. Suppose that Aβα is a submatrix of AHm×n whose rows and columns are indexed by α and β, respectively. Then, Aαα is a principal submatrix of A whose rows and columns are indexed by α. If A is Hermitian, then Aαα stands for a principal minor of detA. The collection of strictly increasing sequences of 1kn integers chosen from 1n is denoted by Lk,nα:α=α1αk1α1<<αkn. For fixed iα and jβ, put Ir,miα:αLr,miα, Jr,njβ:βLr,njβ.

Let a.j and a.j be the jth columns, ai. and ai. be the ith rows of A and A, respectively. Suppose that Ai.b and A.jc stand for the matrices obtained from A by replacing its ith row with the row vector bH1×n and its jth column with the column vector cHm, respectively.

Based on determinantal representations of the Moore–Penrose inverse obtained in [28], we have determinantal representations of the projections.

Lemma 3.2. [28] If AHrm×n, then the determinantal representations of the projection matrices AAQA=qijAn×n and AAPA=pijAm×m can be expressed as follows

qijA=βJr,nicdetiAA.iȧ.jβββJr,nAAββ=αIr,njrdetjAA.jȧ.iαααIr,nAAαα,E7
pijA=αIr,mjrdetjAAj.a¨i.αααIr,mAAαα=βJr,micdetiAA.ia¨.jβββJr,mAAββ,E8

where ȧ.i and ȧ.j, a¨i. and a¨.j are the ith rows and the jth columns of AAHn×n and AAHm×m, respectively.

Recently, D-representations of the quaternion core-EP inverses were obtained in [1] as well.

Lemma 3.3. [1] Suppose that AHn×n, IndA=k and rankAk=s. Then A=aij,r and A=aij,l possess the determinantal representations, respectively,

aij,r=αIs,njrdetjAk+1Ak+1j.âi.αααIs,nAk+1Ak+1αα,E9
aij,l=βJs,nicdetiAk+1Ak+1.ia.jβββJs,nAk+1Ak+1ββ,E10

where âi. is the ith row of Â=AkAk+1 and a.j is the jth column of A=Ak+1Ak.

Theorem 3.4. Let AHsn×n, IndA=k and rankAk=s1. Then its MPCEP inverse A,=aij, is expressed by componentwise

aij,=αIs1,njrdetjAk+1Ak+1j.vi.1ααβJs,nAAββαIs1,nAk+1Ak+1ααE11
=βJs,nicdetiAA.iu.j1βββJs,nAAββαIs1,nAk+1Ak+1αα,E12

where

vi.1=βJs,nicdetiAA.ia˜.lββH1×n,l=1,,n,E13
u.j1=αIs1,njrdetjAk+1Ak+1j.a˜f.ααHn×1,f=1,,n,

and a˜.l and a˜f. are the lth column and the fth row of A˜=AAk+1Ak+1.

Proof. By (2), we have

aij,=l=1nqilalj,r.E14

Using (7) and (9) for the determinantal representations of QA=AA=qij and A, respectively, from (14) it follows

aij,=l=1nf=1nβJs,nicdetiAA.iȧ.fβββJs,nAAββ×αIs1,njrdetjAk+1Ak+1j.âl.αααIs1,nAk+1Ak+1αα==l=1nf=1nβJs,nicdetiAA.ie.fβββJs,nAAββa˜flαIs1,njrdetjAk+1Ak+1j.el.αααIs1,nAk+1Ak+1αα,

where e.f and el. are the fth column and the lth row of the unit matrix In, âl. is the lth row of Â=AkAk+1, and a˜fl is the (fl)th element of A˜=AAk+1Ak+1.

If we denote by

vil1f=1nβJs,nicdetiAA.ie.fββa˜fl=βJs,nicdetiAA.ia˜.lββ

the lth component of a row-vector vi.1=vi11vin1, then

l=1nvil1αIs1,njrdetjAk+1Ak+1j.el.αα=αIs1,njrdetjAk+1Ak+1j.vi.1αα.

So, we have (11). By putting

ufj1l=1na˜flαIs1,njrdetjAk+1Ak+1j.el.αα=αIs1,njrdetjAk+1Ak+1j.a˜f.αα

as the fth component of a column-vector u.j1=u1j1unj1T, it follows

f=1nβJs,nicdetiAA.ie.fββufj1=βJs,nicdetiAA.iu.j1ββ.

Hence, we obtain (12). □

Determinantal representations of a complex MPCEP inverse are obtained by substituting row-column determinants for usual determinants in (11)(12).

Corollary 3.5. Let ACsn×n, IndA=k and rankAk=s1. Then its MPCEP inverse A,=aij, has the following determinantal representations

aij,=αIs1,njAk+1Ak+1j.vi.1ααβJs,nAAββαIs1,nAk+1Ak+1αα=βJs,niAA.iu.j1βββJs,nAAββαIs1,nAk+1Ak+1αα,

where

vi.1=βJs,niAA.ia˜.lββC1×n,l=1,,n,u.j1=αIs1,njAk+1Ak+1j.a˜f.ααCn×1,f=1,,n,E15

and a˜.l and a˜f. are the lth column and the fth row of A˜=AAk+1Ak+1.

Theorem 3.6. Let AHsn×n, IndA=k and rankAk=s1. Then its CEPMP inverse A,=aij, has the following determinantal representations

aij,=αIs,njrdetjAAj.vi.2αααIs,nAAααβJs1,nAk+1Ak+1ββE16
=βJs1,nicdetiAk+1Ak+1.iu.j2ββαIs,nAAααβJs1,nAk+1Ak+1ββ,E17

where

vi.2=βJs1,nicdetiAk+1Ak+1.iâ.lββH1×n,l=1,,n,u.j2=[αIs,njrdetjAAj.(âf.)αα]Hn×1,f=1,,n.E18

Here â.l and âf. are the lth column and the fth row of Â=Ak+1Ak+1A.

Proof. The proof is similar to the proof of Theorem 3.4 by using the representation (3) for the CEPMP inverse.

Corollary 3.7. Let ACsn×n, IndA=k and rankAk=s1. Then its CEPMP inverse A,=aij, has the following determinantal representations

aij,=αIs,njAAj.vi.2αααIs,nAAααβJs1,nAk+1Ak+1ββ=βJs1,nicdetiAk+1Ak+1.iu.j2ββαIs,nAAααβJs1,nAk+1Ak+1ββ,

where

vi.2=βJs1,niAk+1Ak+1.iâ.lββC1×n,l=1,,n,u.j2=αIs,njAAj.âf.ααCn×1,f=1,,n.E19

Here â.l and âf. are the lth column and the fth row of Â=Ak+1Ak+1A.

Theorem 3.8. Let AHsn×n, IndA=k and rankAk=s1. Then its right MPCEPMP inverse A,,,r=aij,,,r has the following determinantal representations

aij,,,r=αIs,njrdetjAAj.ϕi.1αααIs,nAAαα2βJs1,nAk+1Ak+1ββ=E20
=βJs1,nicdetiAk+1Ak+1.iψ.j1βββIs,nAAββ2αJs1,nAk+1Ak+1αα,E21

where

ϕi.1=βJs1,nicdetiAk+1Ak+1.iû.lββH1×n,l=1,,n,ψ.j1=αIs,njrdetjAAj.ûf.ααHn×1,f=1,,n.

Here û.l and ûf. are the lth column and the fth row of Û=U2AA, and the matrix U2 is constructed from the columns (18).

Proof. Owing to (4), we have

aij,,,r=t=1nait,ptj.E22

Applying (8) for the determinantal representation of PA=AA=pij and (17) for the determinantal representation of A, in (22), we obtain

aij,,,r=t=1nβJs1,nicdetiAk+1Ak+1.iu.t2ββαIs,nAAααβIs1,nAk+1Ak+1ββ×αIs,njrdetjAAj.a¨t.αααIs,nAAαα==l=1nf=1nβJs1,nicdetiAk+1Ak+1.ie.fβββJs1,nAk+1Ak+1ββûfl×αIs,njrdetjAAj.el.αααIs,nAAαα2,

where e.f and el. are the fth column and the lth row of the unit matrix In, and ûfl is the (fl)th element of Û=U2AA. The matrix U2=u.12u.n2 is constructed from the columns (18). If we denote by

ϕil1f=1nβJs1,nicdetiAk+1Ak+1.ie.fββûfl=βJs1,nicdetiAk+1Ak+1.iû.lββ

the lth component of a row-vector ϕi.1=ϕi11ϕin1, then

l=1nϕil1αIs,njrdetjAAj.el.αα=αIs,njrdetjAAj.ϕi.1αα.

Therefore, (20) holds.

By putting

ψfj1l=1nâflαIs,njrdetjAAj.el.αα=αIs,njrdetjAAj.(ûf.αα

as the fth component of a column-vector ψ.j1=ψ1j1ψnj1T, it follows

f=1nβJs1,nicdetiAk+1Ak+1.ie.fββψfj1=βJs1,nicdetiAk+1Ak+1.iψ.j1ββ.

Thus, Eq. (21) holds.

Corollary 3.9. Let ACsn×n, IndA=k and rankAk=s1. Then its right MPCEPMP inverse A,,,r=aij,,,r has the following determinantal representations

aij,=αIs,njAAj.ϕi.1αααIs,nAAαα2βJs1,nAk+1Ak+1ββ=βJs1,niAk+1Ak+1.iψ.j1βββIs,nAAββ2αJs1,nAk+1Ak+1αα,

where

ϕi.1=βJs1,niAk+1Ak+1.iû.lββC1×n,l=1,,n,ψ.j1=αIs,njAAj.ûf.ααCn×1,f=1,,n.

Here û.l and ûf. are the lth column and the fth row of Û=U2AA, and the matrix U2 is constructed from the columns (19).

Theorem 3.10. Let AHsn×n, IndA=k and rankAk=s1. Then its left MPCEPMP inverse A,,,l=aij,,,l has the following determinantal representations

aij,,,l=αIs1,njrdetjAk+1Ak+1j.ϕi.2ααβJs,nAAββ2αIs1,nAk+1Ak+1αα=E23
=βJs,nicdetiAA.iψ.j2βββJs,nAAββ2αIs1,nAk+1Ak+1αα,E24

where

ϕi.2=βJs,nicdetiAA.iv˜.tββH1×n,t=1,,n,ψ.j2=αIs1,njrdetjAk+1Ak+1j.v˜f.ααHn×1,f=1,,n,

and v˜.l and v˜f. are the lth column and the fth row of V˜=AAV1, where the matrix V1 is determined from the rows (13).

Proof. Due to (6),

aij,,,l=t=1nqitatj,.E25

Using (7) for the determinantal representation of QA=AA=qij and (9) for the determinantal representation of A, in (14), we obtain

aij,,,l=t=1nβJs,nicdetiAA.iȧ.tβββJs,nAAββ×αIs1,njrdetjAk+1Ak+1j.vt.1ααβJs,nAAββαIs1,nAk+1Ak+1αα=t=1nf=1nβJs,nicdetiAA.ie.fβββJs,nAAββ2v˜ft×αIs1,njrdetjAk+1Ak+1j.et.αααIs1,nAk+1Ak+1αα,

where v˜ft is the (ft)th element of V˜=AAV1 and the matrix V1 is constructed from the rows (13). If we put

ϕit2f=1nβJs,nicdetiAA.ie.fββv˜ft=βJs,nicdetiAA.iv˜.tββ

as the lth component of a row-vector ϕi.2=ϕi12ϕin2, then

t=1nϕit2αIs1,njrdetjAk+1Ak+1j.et.αα=αIs1,njrdetjAk+1Ak+1j.ϕi.2αα,

then Eq. (23) holds. If we denote by

ψfj2t=1nu˜ftαIs1,njrdetjAk+1Ak+1j.et.αα==αIs1,njrdetjAk+1Ak+1j.(u˜f.αα

the fth component of a column-vector ψ.j2=ψ1j2ψnj2T, then

f=1nβJs,nicdetiAA.ie.fββψfj2=βJs,nicdetiAA.iψ.j2ββ.

Hence, we obtain (24). □

Corollary 3.11. Let ACsn×n, IndA=k and rankAk=s1. Then its left MPCEPMP inverse A,,,l=aij,,,l has the following determinantal representations

aij,,,l=αIs1,njAk+1Ak+1j.ϕi.2ααβJs,nAAββ2αIs1,nAk+1Ak+1αα==βJs,niAA.iψ.j2βββJs,nAAββ2αIs1,nAk+1Ak+1αα,

where

ϕi.2=βJs,niAA.iv˜.tββC1×n,t=1,,n,ψ.j2=αIs1,njAk+1Ak+1j.v˜f.ααCn×1,f=1,,n,

and v˜.t and v˜f. are the tth column and the fth row of V˜=AAV1, where the matrix V1 is determined by (13).

## 4. Conclusions

In this chapter, notions of the MPCEP and CEPMP inverses are extended to quaternion matrices, and the new right and left MPCEPMP inverses are introduced and their characterizations are explored. Their determinantal representations are obtained within the framework of the theory of noncommutative column-row determinants previously introduced by the author. Also, determinantal representations of these generalized inverses for complex matrices are derived by using regular determinants. The obtained determinantal representations give new direct methods of calculations of these generalized inverses.

## Acknowledgments

The author thanks the Erwin Schrödinger Institute for Mathematics and Physics (ESI) at the University of Vienna for the support given by the Special Research Fellowship Programme for Ukrainian Scientists.

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Written By

Ivan I. Kyrchei

Reviewed: 07 February 2022 Published: 23 April 2022