# Real Representation Approach to Quaternion Matrix Equation Involvingϕ-Hermicity

Authors: Xin Liu; Huajun Huang; Zhuo-Heng He
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

For a quaternion matrixA, we denote by Aϕ the matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a nonstandard involution of quaternions. A is said to be ϕ-Hermitian or ϕ-skew-Hermitian if A=Aϕ or A=−Aϕ, respectively. In this paper, we give a complete characterization of the nonstandard involutions ϕ of quaternions and their conjugacy properties; then we establish a new real representation of a quaternion matrix. Based on this, we derive some necessary and sufficient conditions for the existence of a ϕ-Hermitian solution or ϕ-skew-Hermitian solution to the quaternion matrix equation AX=B. Moreover, we give solutions of the quaternion equation when it is solvable.


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## Body

## 1. Introduction

The definitions ofϕ-Hermitian and ϕ-skew-Hermitian quaternion matrices were first introduced by Rodman (Definition 3.6.1 in [1]). As a special case of ϕ-(skew)-Hermitian matrix, the η-(anti)-Hermitian matrix first arises in widely linear modelling and has important applications over independent component analysis [2] and convergence analysis in statistical signal processing [3]. Recently, it was widely investigated (e.g., [4–17]). For instance, Horn and Zhang [15] gave a singular value decomposition for η-Hermitian matrix. Yuan et al. [16, 17] derived least squares η-(anti)-Hermitian solution of some classic quaternion matrix equations, while there have been a few papers to consider the ϕ-Hermitian and ϕ-skew-Hermitian solutions of some quaternion matrix equations. For example, the ϕ-Hermitian solutions of the mixed pairs of Sylvester matrix equations,(1)A1X−YB1=C1,A2Z−YB2=C2,Z=Zϕ,A1X−YB1=C1,A2Y−ZB2=C2,Z=Zϕ,were given by using Moore–Penrose inverses [18]. The ϕ-skew-Hermitian solution of the quaternion matrix equations,(2)BXBϕ+CYCϕ=A,EZEϕ+DYDϕ=B,was discussed through matrix decomposition [19]. Very recently, some practical necessary and sufficient conditions for the existence of a solution of the system of quaternion matrix equations(3)A1X1+A1X1ϕ+C1Y1C1ϕ+F1WF1ϕ=E1,A2X2+A2X2ϕ+C2Y2C2ϕ+F2WF2ϕ=E2,Y1=Y1ϕ,Y2=Y2ϕ,W=Wϕ,were given by [20] in terms of ranks and Moore–Penrose inverses. As we know that real (complex) representation is a usual method to address the problems of quaternion matrix theory, it enables us to convert problems over quaternions into the problems over real (complex) number field. Up to now, there are many existing real (complex) representations. But, as far as we know, none of those can well preserve the structure of ϕ-(skew)-Hermitian matrix. Thus, we define a new real representation of a quaternion matrix, which can map a ϕ-Hermitian matrix or ϕ-skew-Hermitian matrix into a skew-symmetric or symmetric matrix. Motivated by the above works and its important applications, we will present a new real representation of a quaternion matrix to discuss the ϕ-Hermitian or ϕ-skew-Hermitian solution of the quaternion matrix equation AX=B.The paper is organized as follows. In Section2, we give a complete characterization of the nonstandard involutions of ℍ and their conjugacy properties. In Section 3, we present a new real representation of quaternion matrix, which could well preserve the structure of ϕ-(skew)-Hermitian matrix. In Section 4, we discuss the existence of the ϕ-(skew)-Hermitian solution of the quaternion matrix equation AX=B and derive the solutions when it is solvable.


## 2. A Complete Characterization of the Nonstandard Involutions ofℍ and Their Conjugacy Properties

In this section, we give a complete characterization of the nonstandard involutions ofℍ and their conjugacy properties.Letℝ and ℍm×n stand, respectively, for the real number field and the set of all m×n matrices over the real quaternion algebra(4)ℍ=a0+a1i+a2j+a3k∣i2=j2=k2=ijk=−1,a0,a1,a2,a3∈ℝ.The symbolsIn,0,AT stand for the n×n identity matrix, the zero matrix with appropriate size, the transpose of A, respectively. The Moore–Penrose inverse A† of a real matrix A is defined to be the unique matrix A†, such that(i)
AA†A=A,(ii)
A†AA†=A†,(iii)
AA†T=AA†,(iv)
A†AT=A†A.Furthermore,LA and RA stand for the projectors LA=I−A†A and RA=I−AA† induced by A, respectively. For any A=A1+A2i+A3j+A4k∈ℍm×n, A∗=A1T−A2Ti−A3Tj−A4Tk is defined as the usual conjugate transpose of A.In [1], Rodman defined the involution over ℍ as follows.Definition 1 (involution) [1].
A mapϕ: ℍ⟶ℍ is called an antiendomorphism if ϕxy=ϕyϕx for all x,y∈ℍ, and ϕx+y=ϕx+ϕy for all x,y∈ℍ. An antiendomorphism ϕ is called an involution if ϕϕx=x for every x∈ℍ.The matrix representations of involutions are given in the following lemma.Lemma 1 [1].
Letϕ be an antiendomorphism of ℍ. Assume that ϕ does not map ℍ into zero. Then, ϕ is one-to-one and onto ℍ; thus, ϕ is in fact an antiautomorphism. Moreover, ϕ is real linear and can be represented as a 4×4 real matrix with respect to the basis 1,i,j,k. Then, ϕ is an involution if and only if(5)ϕ=100T,where either T=−I3 or T is a 3×3 real orthogonal symmetric matrix with eigenvalues of 1,1,−1.By Lemma 1, involutions can be classified into two classes: the standard involution and the nonstandard involution, as defined below.Definition 2 (standard involution) [1].
An involutionϕ is standard if ϕ=100−I3.Definition 3 (nonstandard involution) [1].
An involutionϕ is nonstandard if(6)ϕ=100T,where T is a 3×3 real orthogonal symmetric matrix with eigenvalues of 1,1,−1.
In this paper, we only consider the nonstandard involution. The following theorem shows that each nonstandard involution is in the form ofϕa=aη∗=−ηa∗η, where η=u1i+u2j+u3k∈ℍ is unit and pure imaginary and a∗ stands for the conjugate transpose of the quaternion a.Theorem 1.
A mapϕ:ℍ⟶ℍ is a nonstandard involution if and only if it is of the form ϕa=aη∗=−ηa∗η, where η=u1i+u2j+u3k∈ℍ is unit and pure imaginary, that is, u1,u2,u3∈ℝ and u12+u22+u32=1. Moreover, let u=u1,u2,u3T, then the matrix representation of such a nonstandard involution is ϕ=100T, where T=I3−2uuT is a real Householder matrix.Proof.
First, for eachη=u1i+u2j+u3k∈ℍ with u1,u2,u3∈ℝ and u12+u23+u32=1, we show that ϕa=−ηa∗η defines a nonstandard involution. By direct calculation, we have ϕ1=1, and(7)ϕi=1−2u12i−2u1u2j−2u1u3k,ϕj=−2u2u1i+1−2u22j−2u2u3k,ϕk=−2u3u1i−2u3u2j+1−2u32k.
Hence, the matrix representation ofϕ with respect to 1,i,j,k is ϕ=100T, where(8)T=1−2u12−2u2u1−2u3u1−2u1u21−2u22−2u3u2−2u1u3−2u2u31−2u32=I3−2uuT,u=u1u2u3.
Then,T is a real Householder matrix, which has eigenvalues of 1,1,−1 and is symmetric. Hence, ϕ is a nonstandard involution by Lemma 1.
Second, Lemma 1 shows that every nonstandard involutionϕ has the matrix representation ϕ=100T, where T is an orthogonal matrix with eigenvalues of 1,1,−1. Let e1=1,0,0T. Then, there exists a 3×3 real orthogonal matrix U such that(9)T=Udiag−1,1,1UT=UI3−2e1e1TUT=I3−2Ue1Ue1T,where Ue1 is the first column of U and is a real unit vector. Denote u=Ue1=u1,u2,u3T and η=u1i+u2j+u3k. Then, ϕ is of the form ϕa=−ηa∗η, as desired.We give two examples to understand the nonstandard involution.Example 1 [4].
Leta=a0+a1i+a2j+a3k∈ℍ and η∈i,j,k. Then, the map a⟶aη∗ is a nonstandard involution, where aη∗=−ηa∗η. Upon computation, we have(10)ai∗=a0−a1i+a2j+a3k,ifη=i,aj∗=a0+a1i−a2j+a3k,ifη=j,ak∗=a0+a1i+a2j−a3k,ifη=k.Example 2 [18].
Leta=a0+a1i+a2j+a3k∈ℍ and η∈2/2i+j,2/2i+k,2/2j+k. Then, the map a⟶aη∗ is a nonstandard involution, where aη∗=−ηa∗η. Upon computation, we have(11)a2/2i+j∗=a0−a2i−a1j+a3k,ifη=22i+j,a2/2i+k∗=a0−a3i+a2j−a1k,ifη=22i+k,a2/2j+k∗=a0+a1i−a3j−a2k,ifη=22j+k.
The product of two nonstandard involutions is in general not an involution. However, every product of the formδϕδ−1=δϕδ, where δ and ϕ are nonstandard involutions, is again a nonstandard involution. Moreover, Theorem 1 implies that all nonstandard involutions are conjugate via nonstandard involutions, as shown below.Theorem 2.
Every nonstandard involutionϕ of ℍ is conjugate via a nonstandard involution to ϕi where ϕia=ai∗=−ia∗i. In other words, there exists a nonstandard involution δ such that(12)ϕ=δϕiδ−1=δϕiδ.Proof.
The matrix representation ofϕi is ϕi=diag1,−1,1,1=100I3−2e1e1T. By Theorem 1, suppose the nonstandard involution ϕ has the matrix representation ϕ=100I3−2uuT, where u=u1,u2,u3T is a real unit vector. We hope to find a real Householder matrix V=I3−2vvT, where v=v1,v2,v3T is a real unit vector, such that(13)I3−2uuT=VI3−2e1e1TVT=I3−2Ve1Ve1T.
In other words, we needI3−2vvTe1=Ve1=u, explicitly,(14a)1−2v12=u1,(14b)−2v2v1=u2,(14c)−2v3v1=u3.
Ifu1=1, then u2=u3=0, and (14a) gives v1=0; we simply choose v2,v3∈ℝ such that v22+v32=1. Otherwise u1<1, we may choose v1=1−u1/2≠0 to satisfy (14a) and then solve v2 and v3 by (14b) and (14c). The equations (14a)–(14c) imply that(15)1=u12+u22+u32=1−2v122+−2v2v12+−2v3v12=1+4v12−1+v12+v22+v32.
In both cases above, we will havev12+v22+v32=1. Now let δ be the nonstandard involution with the matrix representation δ=100V. Then, (12) holds.For a given nonstandard involutionϕ over ℍ and A∈ℍm×n, Rodman [1] denotes by Aϕ the n×m matrix obtained by applying ϕ entrywise to the transposed matrix AT. Then, by Theorem 1, the general form of Aϕ follows.Lemma 2.
Letϕ be a nonstandard involution. Then, there exists a unit pure imaginary quaternion η such that Aϕ is in the form of(16)Aϕ=−ηA∗η.
We make an example in the following.Example 3.
IfAϕ=−jA∗j for A∈ℍm×n, then(17)ii+kj−ki+j+k2−i2+ii+2j3j+kϕ=i2−ii+k2+i−j−ki−2ji−j+k−3j+k.
The following are the definitions ofϕ-(skew)-Hermitian matrix. They are the special cases of ϕ-(skew)-Hermitian.Definition 4 (ϕ-Hermitian or ϕ-skew-Hermitian) [1].
A
∈
ℍ
n
×
n is said to be ϕ-Hermitian or ϕ-skew-Hermitian if A=Aϕ or A=−Aϕ, where ϕ is a nonstandard involution.
From Lemma 2 and Definition 4, there follows a equivalent description of Definition 4: letϕ is a nonstandard involution, that is, ϕa=−ηa∗η, where η∈ℍ is unit and pure imaginary. Then, A∈ℍn×n is said to be ϕ-Hermitian or ϕ-skew-Hermitian if A=−ηA∗η or A=ηA∗η. It will play important role in solving our ϕ-Hermicity problem.Example 4.
Forη∈i,j,k, A∈ℍn×n is said to be η-Hermitian or η-anti-Hermitian if A=Aη∗ or A=−Aη∗, where Aη∗=−ηA∗η. Obviously, η-Hermitian (η-anti-Hermitian) matrix is a special case of ϕ-Hermitian (ϕ-skew-Hermitian) matrix.


## 3. New Real Representation of Quaternion Matrix

Real representation method is a powerful tool to study the quaternions. There are several existing real (complex) representations of quaternion matrix, for example, the following real representation in [21, 22].Definition 5.
ForX∈ℍm×n, X=X0+X1i+X2j+X3k,X0,X1,X2,X3∈ℝm×n, define(18)Xσ=X0−X1−X2−X3X1X0−X3X2X2X3X0−X1X3−X2X1X0∈ℝ4m×4n.
The above real representation of a quaternion matrix can map an Hermitian matrix or a skew-Hermitian matrix overℍ into a symmetric matrix or skew-symmetric matrix over ℝ. In this section, we aim at exploring the ϕ-(skew)-Hermitian solutions of the quaternion matrix AX=B. To well preserve the structure of ϕ-(skew)-Hermitian matrix, in this section, we present a new representation of quaternion matrix basing on the above one. Let η=u1i+u2j+u3k∈ℍ be a unit and pure imaginary quaternion. Now, applying the product rule XYσ=XσYσ to the matrix equality(19)−η2In=In,we have(20)−Fn2=I4n,where(21)Fn=ηInσ=0−u1In−u2In−u3Inu1In0−u3Inu2Inu2Inu3In0−u1Inu3In−u2Inu1In0.
Note thatFnT=−Fn and −Fn2=I4n, thus Fn is an orthogonal matrix. Next, we can use Fn to define the new real representation.Definition 6.
ForX=X0+X1i+X2j+X3k∈ℍm×n,X0,X1,X2,X3∈ℝm×n and η=u1i+u2j+u3k∈ℍ, which is unit and pure imaginary. We define(22)Xση=FmXσ.
It follows from Theorem 1 and Lemma 2 that the real representationση can map a ϕ-Hermitian matrix or ϕ-skew-Hermitian matrix over ℍ into a skew-symmetric or symmetric matrix over ℝ. This will be a critical technique in tackling the problem of the paper.
Denoting(23)Pn=0−In00In000000In00−In0,Ln=00−In0000−InIn0000In00,Tn=000−In00In00−In00In000.
Then, by direct calculation, we have the following lemma.Lemma 3.
P
n
,
L
n
,
T
n
,
F
n are all orthogonal matrices, and(24)PnFn=FnPn,LnFn=FnLn,TnFn=FnTn.Next, we summarize the properties of real representationsXσ,Xση in the following proposition. Some properties of σ are given by [21, 22] and the others can be verified directly.Proposition 1.
LetA,B∈ℍm×n,C∈ℍn×s,b∈ℝ. Then(a)
A+Bσ=Aσ+Bσ,bAσ=bAσ,A+Bση=Aση+Bση,bAση=bAση;(b)
ACσ=AσCσ,ACση=−AσηFnCση;(c)
(i)PmTAσPn=Aσ,LmTAσLn=Aσ,TmTAσTn=Aσ; (ii) PmTAσηPn=Aση,LmTAσηLn=Aση,TmTAσηTn=Aση;(d)
A∗σ=AσT,Aη∗ση=−AσηT;(e)
Aση=FmAσ,(f)
A=1/4ImiImjImkImAσIn−iIn−jIn−kIn.
If takingη∈i,j,k, we get real representations for η-(anti)-Hermitian matrices. Those real representations can map an η-Hermitian matrix or η-skew-Hermitian matrix over ℍ into a skew-symmetric or symmetric matrix over ℝ. They could be useful in the studying of η-(anti)-Hermitian matrix.Definition 7.
ForX∈ℍm×n, X=X0+X1i+X2j+X3k,X0,X1,X2,X3∈ℝm×n, define(25)Xσi=GmXτ=−X1−X0X3−X2X0−X1−X2−X3−X3X2−X1−X0X2X3X0−X1∈ℝ4m×4n,whereGm=0−Im00Im000000−Im00Im0,Xσj=HmXτ=−X2−X3−X0X1X3−X2X1X0X0−X1−X2−X3−X1−X0X3−X2∈ℝ4m×4n,whereHm=00−Im0000ImIm0000−Im00,Xσk=KmXτ=−X3X2−X1−X0−X2−X3−X0X1X1X0−X3X2X0−X1−X2−X3∈ℝ4m×4n,whereKn=000−Im00−Im00Im00Im000.
As corollary of Proposition 1, we have:Proposition 2.
LetA,B∈ℍm×n,C∈ℍn×s,b∈ℝ,η∈i,j,k. Then(a)
A+Bση=Aση+Bση,bAση=bAση;(b)
ACσi=−AσiGnCσi,ACσj=−AσjHnCσj,ACσk=−AσkKnCσk;(c)
PmTAσηPn=Aση,LmTAσηLn=Aση,TmTAσηTn=Aση;(d)
Aη∗ση=−AσηT;(e)
Aσi=GmAσ,Aσj=HmAσ,Aσk=KmAσ.


## 4.ϕ-Hermitian or ϕ-Skew-Hermitian Solution to AX = B

In this section, we apply some new real representations given above to discuss the solvability conditions andϕ-Hermitian or ϕ-skew-Hermitian solution of the quaternion matrix equation AX=B. We first begin with a useful lemma, which can be slightly modified from ℂ to ℝ.Lemma 4 [23].
LetA,B∈ℝm×n. Then, the matrix equation AX=B has a solution X=±XT∈ℝn×n if and only if RAB=0 and BAT=±ABT. In which case, the general X=±XT solution to AX=B is(26)X=A†B±A†BT∓A†ABTA†T+LAULAT,where U=±UT∈ℝn×n is an arbitrary matrix.Applying Lemma 4 and real representationsση,σ, we get the following statements.Theorem 3.
LetXϕ=−ηX∗η be given, where η is a unit and pure imaginary quaternion. Then, the quaternion matrix equation(27)AX=B,has a ϕ-Hermitian (ϕ-skew-Hermitian) solution X=Xϕ∈ℍn×n (X=−Xϕ∈ℍn×n) if and only if one of the following equivalent statements hold:(a)
The real matrix equation(28)AσηY=BσηFn,has a real solutionY=−YT∈ℝ4n×4n (Y=YT∈ℝ4n×4n).(29)bRAσηBσηFn=0,BσηFnAσηT=−AσηBσηFnT,orRAσηBσηFn=0,BσηFnAσηT=AσηBσηFnT.Moreover, if (27) is consistent, then(30)X=−Zη,is aX=±Xη∗ solution to (27), where(31)Z=116IniInjInkInY+PnYPnT+LnYLnT+TnYTnTIn−iIn−jIn−kIn,with(32)Y=Aση†BσηFn∓Aση†BσηFnT±Aση†AσηBσηFnTAση†T+LAσηULAσηT,andU=∓UT∈ℝ4n×4n is arbitrary.Proof.
We only prove the case ofX=Xϕ. Suppose that X is a ϕ-Hermitian solution of (27), then it follows from (b) and (d) of Proposition 2 that −AσηFnXση=Bση and Xση=−XσηT. Postmultiplying the both sides with Fn gives(33)−AσηFnXσηFn=BσηFn.
SinceFnT=−Fn, then (33) can be rewritten as AσηFnXσηFnT=BσηFn. Hence, FnXσηFnT is a skew-symmetric solution to (28).
Conversely, if (28) has a skew-symmetric solution Y, i.e., AσηY=BσηFn,Y=−YT. Then, using (ii) in (c) of Proposition 1 gives(34)PmTAσηPnY=PmTBσηPnFn,LmTAσηLnY=LmTBσηLnFn,TmTAσηTnY=TmTBσηTnFn.
Note thatPnFn=FnPn,LnFn=FnLn,TnFn=FnTn. Hence,(35)PmTAσηPnY=PmTBσηFnPn,LmTAσηLnY=LmTBσηFnLn,TmTAσηTnY=TmTBσηFnTn.
Then, by Lemma 3,(36)AσηPnYPnT=BσηFn,AσηLnYLnT=BσηFn,AσηTnYTnT=BσηFn,which follows that PnYPnT,LnYLnT,TnYTnT are skew-symmetric solutions of (28). Then, so is Y≔1/4Y+PnYPnT+LnYLnT+TnYTnT. For the given skew-symmetric solution Y, we set(37)Y=Y11Y12Y13Y14Y21Y22Y23Y24Y31Y32Y33Y34Y41Y42Y43Y44.
By direct computation(38)Y=Z1−Z2−Z3−Z4Z2Z1−Z4Z3Z3Z4Z1−Z2Z4−Z3Z2Z1,where(39)Z1=14Y11+Y22+Y33+Y44,Z2=14Y21−Y12+Y43−Y34,Z3=14Y31−Y42−Y13+Y24,Z4=14Y41+Y32−Y23−Y14.
Now, construct a quaternion matrix(40)Z=Z1+Z2i+Z3j+Z4k=116ImiImjImkImY+PmYPnT+LmYLnT+TmYTnTIn−iIn−jIn−kIn.
It can be seen from (f) of Proposition 1 that Zσ=Y. Y is a skew-symmetric solution of (28); thus, Zσ=−ZσT and AσηZσ=BσηFn. Put X=−Zη. η is unit and pure imaginary; thus, −η=η−1. Then, Xη=Z. Observing that(41)FnXσηFnT=−FnXσηFn=XσFn=XσηInσ=Xησ=Zσ.
Thus, substituting it intoAσηZσ=BσηFn gives AσηFnXσηFnT=BσηFn. As we recall that it is equivalent to AX=B. Since Zσ=−ZσT, Xση=−XσηT. By Lemma 2 and (d) of Proposition 1, X=Xη∗=−ηX∗η=Xϕ. Therefore, X is a ϕ-Hermitian solution of (27). That is to say, once the skew-symmetric solution Y to (28) is given, then the blocks Yij can be read directly from Y. Then, Z can be determined immediately by (40). Thus, our required quaternion solution X follows by taking Z into the formula X=−Zη. By Lemma 4, Y given by (32) is a skew-symmetric solution to (28), and thus X in (30) is a ϕ-Hermitian solution to (27).
At the end, we can see that (27) has a X=Xϕ solution if and only if its real matrix equation (28) has a skew-symmetric solution.Remark 1.
If we setη∈i,j,k, then ϕ-Hermitian or ϕ-skew-Hermitian matrix is reduced to the well-known η-Hermitian or η-anti-Hermitian matrix. Thus, the η-Hermitian or η-anti-Hermitian solution to (27) can be discussed by simply replacing Fn by Gn,Hn,Kn and ση by σi,σj,σk, respectively.Example 5.
Find aϕ-skew-Hermitian solution to the quaternion matrix equation AX=B, where Xϕ=−2/2i+jX∗2/2i+j,(42)A=23−21+0−211i+32−32j+4−2−11k,B=−8−5−14+3−4−4−3i+38−2−9j+−8712k.
By Theorem 3, its corresponding real matrix equation is(43)AσηY=BσηF2,whereη=22i+j.
SinceRAσηBσηF2=0 and BσηF2AσηT=AσηBσηF2T. Thus, the real equation has a symmetric solution(44)Y=2−1−1000−100−1100100000−1−1000100−1100−100100−1−100−1000−1100000−100−1−1001000−11.
By the formula ofX in (30) in Theorem 3, the quaternion matrix equation also has a solution(45)X=01−10+111−1i+111−1j+01−10k,which one can verify that X=−Xϕ.


## 5. Conclusion

In this paper, we give a complete characterization of the nonstandard involutionsϕ of ℍ and their conjugacy properties. Basing on the characterization of the nonstandard involutions, we present a new real representation of a quaternion matrix, which maps a ϕ-Hermitian or ϕ-skew-Hermitian quaternion matrix into a skew-symmetric or symmetric real matrix. By using this approach, we derive the necessary and sufficient conditions for the existence of a ϕ-Hermitian solution or ϕ-skew-Hermitian solution of the quaternion matrix equation (25). Furthermore, we get solutions of (25) when it is solvable. Moreover, Example 5 is presented to illustrate our results.



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