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Quaternion MPCEP, CEPMP, and MPCEPMP Generalized Inverses

Written By

Ivan I. Kyrchei

Reviewed: February 7th, 2022 Published: April 23rd, 2022

DOI: 10.5772/intechopen.103087

IntechOpen
Matrix Theory - Classics and Advances Edited by Mykhaylo Andriychuk

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Matrix Theory - Classics and Advances [Working Title]

Dr. Mykhaylo I. 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×nmatrices over the quaternion skew field

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

is represented by Hm×n, while Hrm×nis reserved for the subset of Hm×nwith 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 rankAand A, respectively. A matrix AHn×nis said to be Hermitian if A=A. Also,

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

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

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

  • NlA=dH1×m:dA=0is 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 ofAHn×mis the unique matrixA=Xdetermined by equations

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

Definition 1.2.The Drazin inverse ofAHn×nis the uniqueAd=Xthat satisfyingEq.(2) from(1) and the following equations,

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

where k=IndAis 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 Asatisfying the conditions i,j,is called an ij-inverse of A, and is denoted by Aij. In particular, A1is called the inner inverse, A2is called the outer inverse, and A12is called the reflexive inverse, and A1,2,3,4is the Moore–Penrose inverse, etc.

Note that the Moore–Penrose inverse inducts the orthogonal projectors PA=AAand QA=AAonto the right column spaces of Aand 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 ofAHn×nis the unique matrixA=Xwhich satisfies

X=XAX,CrX=CrAd=CrX.

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 ofAHn×nis the unique matrixA=Xfor which

X=XAX,RlX=RlAd=RlX.

Recall that, A=AmAmAdfor mIndA.

Since the quaternion core-EP inverse Ais related to the right space CrAof AHn×nand the quaternion dual core-EP inverse Ais 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.

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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.LetAHn×n. The MPCEP (or MP-Core-EP) inverse ofAis the unique solutionA,=Xto the system

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

The CEPMP (or Core-EP-MP) inverse of Ais the unique solution A,=Xto 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.SupposeAHn×n. The right MPCEPMP inverse ofAis defined as

A,,,r=AAAAA.

The left MPCEPMP inverse of Ais defined as

A,,,l=AAAAA.

The following gives the characteristic equations of these generalized inverses.

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

  1. Xis the right MPCEPMP inverse ofA.

  2. X=A,PA.E4

  3. Xis 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 Xand X1are two solutions of this system. Then XA=A,A=X1Aand AX=AAPA=AX1, which give XAX=XAX1=X1AX1=X1. Therefore, Xis the unique solution to the system. □

The next theorem can be proved in the same way.

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

  1. Xis the left MPCEPMP inverse ofA.

  2. X=QAA,.E6

  3. Xis the unique solution to the system:

1.X=XAX,2.AX=AA,,3.XA=AA.
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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 nrow determinants (-determinants) and ncolumn determinants (-determinants) by stating a certain order of factors in each term.

Definition 3.1.LetA=aijHn×n.

  • For an arbitrary row indexiIn, the ith-determinant ofAis defined as

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

in which Sndenotes the symmetric group on In=1n, while the permutation σis 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 indexjIn, the jth-determinant ofAis defined as the sum

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

in which a permutationτis 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 Athat 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 αα1αk1mand ββ1βk1nbe subsets with 1kminmn. Suppose that Aβαis a submatrix of AHm×nwhose rows and columns are indexed by αand β, respectively. Then, Aααis a principal submatrix of Awhose rows and columns are indexed by α. If Ais Hermitian, then Aααstands for a principal minor of detA. The collection of strictly increasing sequences of 1knintegers chosen from 1nis denoted by Lk,nα:α=α1αk1α1<<αkn.For fixed iαand jβ, put Ir,miα:αLr,miα, Jr,njβ:βLr,njβ.

Let a.jand a.jbe the jth columns, ai.and ai.be the ith rows of Aand A, respectively. Suppose that Ai.band A.jcstand for the matrices obtained from Aby replacing its ith row with the row vector bH1×nand 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] IfAHrm×n, then the determinantal representations of the projection matricesAAQA=qijAn×nandAAPA=pijAm×mcan 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.iand ȧ.j, a¨i.and a¨.jare the ith rows and the jth columns of AAHn×nand AAHm×m, respectively.

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

Lemma 3.3.[1] Suppose thatAHn×n, IndA=kandrankAk=s. ThenA=aij,randA=aij,lpossess 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+1and a.jis the jth column of A=Ak+1Ak.

Theorem 3.4.LetAHsn×n, IndA=kandrankAk=s1. Then its MPCEP inverseA,=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˜.land 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=qijand 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.fand 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˜flis 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.LetACsn×n, IndA=kandrankAk=s1. Then its MPCEP inverseA,=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

anda˜.landa˜f.are the lth column and the fth row ofA˜=AAk+1Ak+1.

Theorem 3.6.LetAHsn×n, IndA=kandrankAk=s1. Then its CEPMP inverseA,=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.land 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.LetACsn×n, IndA=kandrankAk=s1. Then its CEPMP inverseA,=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.land âf.are the lth column and the fth row of Â=Ak+1Ak+1A.

Theorem 3.8.LetAHsn×n, IndA=kandrankAk=s1. Then its right MPCEPMP inverseA,,,r=aij,,,rhas 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.

Hereû.landûf.are the lth column and the fth row ofÛ=U2AA, and the matrixU2is 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=pijand (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.fand el.are the fth column and the lth row of the unit matrix In, and ûflis the (fl)th element of Û=U2AA. The matrix U2=u.12u.n2is 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.LetACsn×n, IndA=kandrankAk=s1. Then its right MPCEPMP inverseA,,,r=aij,,,rhas 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.

Hereû.landûf.are the lth column and the fth row ofÛ=U2AA, and the matrixU2is constructed from the columns(19).

Theorem 3.10.LetAHsn×n, IndA=kandrankAk=s1. Then its left MPCEPMP inverseA,,,l=aij,,,lhas 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,

andv˜.landv˜f.are the lth column and the fth row ofV˜=AAV1, where the matrixV1is determined from the rows(13).

Proof.Due to (6),

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

Using (7) for the determinantal representation of QA=AA=qijand (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˜ftis the (ft)th element of V˜=AAV1and the matrix V1is 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.LetACsn×n, IndA=kandrankAk=s1. Then its left MPCEPMP inverseA,,,l=aij,,,lhas 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,

andv˜.tandv˜f.are the tth column and the fth row ofV˜=AAV1, where the matrixV1is determined by(13).

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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.

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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: February 7th, 2022 Published: April 23rd, 2022