# The Real and Complex Hermitian Solutions to a System of Quaternion Matrix Equations with Applications

Authors: Shao-Wen Yu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We establish necessary and sufficient conditions for the existence of and the expressions for the general real and complex Hermitian solutions to the classical system of quaternion matrix equationsA1X=C1,XB1=C2, andA3XA3*=C3. Moreover, formulas of the maximal and minimal ranks of four real matrices X1,X2,X3, and X4 in solution X=X1+X2i+X3j+X4k to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations A1X=C1,XB1=C2,A3XA3*=C3, andA4XA4*=C4 to have real and complex Hermitian solutions.


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## Body

## 1. Introduction

Throughout this paper, we denote the real number field byℝ; the complex field by ℂ; the set of all m×n matrices over the quaternion algebra
(1.1)H={a0+a1i+a2j+a3k∣i2=j2=k2=ijk=-1,a0,a1,a2,a3∈R}
by ℍm×n; the identity matrix with the appropriate size by I; the transpose, the conjugate transpose, the column right space, the row left space of a matrix A over ℍ by AT,A*,ℛ(A), 𝒩(A), respectively; the dimension of ℛ(A) by dimℛ(A). By [1], for a quaternion matrix A,dimℛ(A)=dim𝒩(A). dimℛ(A) is called the rank of a quaternion matrix A and denoted by r(A). The Moore-Penrose inverse of matrix A over ℍ by A† which satisfies four Penrose equations AA†A=A,A†AA†=A†,(AA†)*=AA†, and (A†A)*=A†A. In this case, A† is unique and (A†)*=(A*)†. Moreover, RA and LA stand for the two projectors LA=I-A†A, and RA=I-AA† induced by A. Clearly, RA and LA are idempotent and satisfies (RA)*=RA,(LA)*=LA,RA=LA*, and RA*=LA.Hermitian solutions to some matrix equations were investigated by many authors. In 1976, Khatri and Mitra [2] gave necessary and sufficient conditions for the existence of the Hermitian solutions to the matrix equations AX=B,AXB=C and
(1.2)A1X=C1,XB2=C2,
over the complex field ℂ, and presented explicit expressions for the general Hermitian solutions to them by generalized inverses when the solvability conditions were satisfied. Matrix equation that has symmetric patterns with Hermitian solutions appears in some application areas, such as vibration theory, statistics, and optimal control theory ([3–7]). Groß in [8], and Liu et al. in [9] gave the solvability conditions for Hermitian solution and its expressions of
(1.3)AXA*=B
over ℂ in terms of generalized inverses, respectively. In [10], Tian and Liu established the solvability conditions for
(1.4)A3XA3*=C3,A4XA4*=C4
to have a common Hermitian solution over ℂ by the ranks of coefficient matrices. In [11], Tian derived the general common Hermitian solution of (1.4). Wang and Wu in [12] gave some necessary and sufficient conditions for the existence of the common Hermitian solution to equations
(1.5)A1X=C1,XB2=C2,A3XA3*=C3,(1.6)A1X=C1,XB2=C2,A3XA3*=C3,A4XA4*=C4,
for operators between Hilbert C*-modules by generalized inverses and range inclusion of matrices.As is known to us, extremal ranks of some matrix expressions can be used to characterize nonsingularity, rank invariance, range inclusion of the corresponding matrix expressions, as well as solvability conditions of matrix equations ([4, 7, 9–24]). Real matrices and its extremal ranks in solutions to some complex matrix equation have been investigated by Tian and Liu ([9, 13–15]). Tian [13] gave the maximal and minimal ranks of two real matrices X0 and X1 in solution X=X0+iX1 to AXB=C over ℂ with its applications. Liu et al. [9] derived the maximal and minimal ranks of the two real matrices X0 and X1 in a Hermitian solution X=X0+iX1 of (1.3), where B*=B. In order to investigate the real and complex solutions to quaternion matrix equations, Wang and his partners have been studying the real matrices in solutions to some quaternion matrix equations such as AXB=C,
(1.7)A1XB1=C1,A2XB2=C2,AXA*+BXB*=C,
recently ([24–27]). To our knowledge, the necessary and sufficient conditions for (1.5) over ℍ to have the real and complex Hermitian solutions have not been given so far. Motivated by the work mentioned above, we in this paper investigate the real and complex Hermitian solutions to system (1.5) over ℍ and its applications.This paper is organized as follows. In Section2, we first derive formulas of extremal ranks of four real matrices X1,X2,X3, and X4 in quaternion solution X=X1+X2i+X3j+X4k to (1.5) over ℍ, then give necessary and sufficient conditions for (1.5) over ℍ to have real and complex solutions as well as the expressions of the real and complex solutions. As applications, we in Section 3 establish necessary and sufficient conditions for (1.6) over ℍ to have real and complex solutions.


## 2. The Real and Complex Hermitian Solutions to System (1.5) Over ℍ

In this section, we first give a solvability condition and an expression of the general Hermitian solution to (1.5) over ℍ, then consider the maximal and minimal ranks of four real matrices X1,X2,X3, and X4 in solution X=X1+X2i+X3j+X4k to (1.5) over ℍ, last, investigate the real and complex Hermitian solutions to (1.5) over ℍ.For an arbitrary matrixMt=Mt1+Mt2i+Mt3j+Mt4k∈ℍm×n where Mt1,Mt2,Mt3, and Mt4 are real matrices, we define a map ϕ(·) from ℍm×n to ℝ4m×4n by
(2.1)ϕ(Mt)=[Mt1Mt2Mt3Mt4-Mt2Mt1Mt4-Mt3-Mt3-Mt4Mt1Mt2-Mt4Mt3-Mt2Mt1].
By (2.1), it is easy to verify that ϕ(·) satisfies the following properties.(a)
M=N⇔ϕ(M)=ϕ(N).(b)
ϕ(kM+lN)=kϕ(M)+lϕ(N),ϕ(MN)=ϕ(M)ϕ(N),k,l∈ℝ.(c)
ϕ(M*)=ϕT(M),
ϕ(M†)=ϕ†(M).(d)
ϕ(M)=Tm-1ϕ(M)Tn=Rm-1ϕ(M)Rn=Sm-1ϕ(M)Sn, where t=m,n,
(2.2)Tt=[0-It00It000000It00-It0],Rt=[00-It0000-ItIt0000It00],St=[000-It00It00-It00It000].(e)
r[ϕ(M)]=4r(M).(f)
M*=M⇔ϕT(M)=ϕ(M),
M*=-M⇔ϕT(M)=-ϕ(M).The following lemmas provide us with some useful results overℂ, which can be generalized to ℍ.Lemma 2.1 (see [2, Lemma  2.1]).
LetA∈ℍm×n,B=B*∈ℍm×m be known, X∈ℍn×n unknown; then the system (1.3) has a Hermitian solution if and only if
(2.3)AA†B=B.
In that case, the general Hermitian solution of (1.3) can be expressed as
(2.4)X=A†B(A†)*+LAV+V*LA,
where V is arbitrary matrix over ℍ with compatible size.Lemma 2.2 (see [12, Corollary  3.4]).
LetA1,C1∈ℍm×n;B1,C2∈ℍn×s;A3∈ℍr×n;C3∈ℍr×r be known, X∈ℍn×n unknown, and F=B1*LA1,M=SLF,S=A3LA1,D=C2*-B1*A1†C1,J=A1†C1+F†D,G=C3-A3(J+LA1LF*J*)A3*,C3=C3*; then the system (1.5) have a Hermitian solution if and only if
(2.5)A1C2=C1B1,A1C1*=C1A1*,B1*C2=C2*B1,RA1C1=0,RFD=0,RMG=0.
In that case, the general Hermitian solution of (1.5) can be expressed as
(2.6)X=J+LA1LFJ*+LA1LFM†G(M†)*LFLA1+LA1LFLMVLFLA1+LA1LFV*LMLFLA1,
where Vis arbitrary matrix over ℍ with compatible size.Lemma 2.3 (see [21, Lemma  2.4]).
LetA∈ℍm×n,B∈ℍm×k,C∈ℍl×n,D∈ℍj×k, and E∈ℍl×i. Then they satisfy the following rank equalities.(a)
r(CLA)=r[AC]-r(A).(b)
r[BALC]=r[BA0C]-r(C).(c)
r[CRBA]=r[C0AB]-r(B).(d)
r[ABLDREC0]=r[AB0C0E0D0]-r(D)-r(E).Lemma2.3 plays an important role in simplifying ranks of various block matrices.Lemma 2.4 (see [11, Theorem  4.1, Corollary  4.2]).
LetA=±A*∈ℍm×m,B∈ℍm×n, andC∈ℍp×m be given; then
(2.7)max⁡X∈Hn×pr[A-BXC∓(BXC)*]=min⁡{r[ABC*],r[ABB*0],r[AC*C0]},min⁡X∈Hn×pr[A-BXC∓(BXC)*]=2r[ABC*]+max⁡{s1,s2},
where
(2.8)s1=r[ABB*0]-2r[ABC*B*00],s2=r[AC*C0]-2r[ABC*C00].
If ℛ(B)⊆ℛ(C*),
(2.9)max⁡X∈Hn×pr[A-BXC-(BXC)*]=min⁡{r[AC*],r[ABB*0]},(2.10)min⁡X∈Hn×pr[A-BXC-(BXC)*]=2r[AC*]+r[ABB*0]-2r[ABC0].Lemma 2.5 (see [28, Theorem  3.1]).
LetA∈ℍm×n,B1∈ℍm×p1,B3∈ℍm×p3,B4∈ℍm×p4,C2∈ℍq2×n,C3∈ℍq3×n,  and C4∈ℍq4×n be given. Then the matrix equation
(2.11)B1X1+X2C2+B3X3C3+B4X4C4=A
is consistent if and only if
(2.12)r[AB1C20C30C40]=r[0B1C20C30C40],r[AB1B3B4C2000]=r[0B1B3B4C2000],r[AB1B3C200C400]=r[0B1B3C200C400],r[AB1B4C200C300]=r[0B1B4C200C300].Theorem 2.6.
System (1.5) has a Hermitian solution over ℍ if and only if the system of matrix equations
(2.13)ϕ(A1)(Yij)4×4=ϕ(C1),(Yij)4×4ϕ(B1)=ϕ(C2),ϕ(A3)(Yij)4×4ϕT(A3)=ϕ(C3),i,j=1,2,3,4,
has a symmetric solution over ℝ. In that case, the general Hermitian solution of (1.5) over ℍ can be written as
(2.14)X=X1+X2i+X3j+X4k=14(Y11+Y22+Y33+Y44)+14(Y12-Y12T+Y34-Y34T)i+14(Y13-Y13T+Y24T-Y24)j+14(Y14-Y14T+Y23-Y23T)k,
where Ytt=YttT;t=1,2,3,4;Y1jT=Yj1;j=2,3,4;Y2jT=Yj2;j=3,4;Y34T=Y43 are the general solutions of (2.13) over ℝ. Written in an explicit form, X1,X2,X3, and X4 in (2.14) are
(2.15)X1=14P1ϕ(X0)P1T+14P2ϕ(X0)P2T+14P3ϕ(X0)P3T+14P4ϕ(X0)P4T+[P1,P2,P3,P4]Lϕ(A1)Lϕ(F)Lϕ(M)V[Lϕ(F)Lϕ(A1)P1TLϕ(F)Lϕ(A1)P2TLϕ(F)Lϕ(A1)P3TLϕ(F)Lϕ(A1)P4T]+[P1,P2,P3,P4]Lϕ(A1)Lϕ(F)VT[Lϕ(M)Lϕ(F)Lϕ(A1)P1TLϕ(M)Lϕ(F)Lϕ(A1)P2TLϕ(M)Lϕ(F)Lϕ(A1)P3TLϕ(M)Lϕ(F)Lϕ(A1)P4T],(2.16)X2=14P1ϕ(X0)P2T-14P2ϕ(X0)P1T+14P3ϕ(X0)P4T-14P4ϕ(X0)P3T+[P1,-P2,P3,-P4]Lϕ(A1)Lϕ(F)Lϕ(M)V[Lϕ(F)Lϕ(A1)P2TLϕ(F)Lϕ(A1)P1TLϕ(F)Lϕ(A1)P4TLϕ(F)Lϕ(A1)P3T]+[P2,P1,P4,P3]Lϕ(A1)Lϕ(F)VT[Lϕ(M)Lϕ(F)Lϕ(A1)P1T-Lϕ(M)Lϕ(F)Lϕ(A1)P2TLϕ(M)Lϕ(F)Lϕ(A1)P3T-Lϕ(M)Lϕ(F)Lϕ(A1)P4T],(2.17)X3=14P1ϕ(X0)P3T-14P3ϕ(X0)P1T+14P4ϕ(X0)P2T-14P2ϕ(X0)P4T+[P1,-P3,P4,-P2]Lϕ(A1)Lϕ(F)Lϕ(M)V[Lϕ(F)Lϕ(A1)P3TLϕ(F)Lϕ(A1)P1TLϕ(F)Lϕ(A1)P2TLϕ(F)Lϕ(A1)P4T]+[P3,P1,P2,P4]Lϕ(A1)Lϕ(F)VT[Lϕ(M)Lϕ(F)Lϕ(A1)P1T-Lϕ(M)Lϕ(F)Lϕ(A1)P3TLϕ(M)Lϕ(F)Lϕ(A1)P4T-Lϕ(M)Lϕ(F)Lϕ(A1)P2T],(2.18)X4=14P1ϕ(X0)P4T-14P4ϕ(X0)P1T+14P2ϕ(X0)P3T-14P3ϕ(X0)P2T+[P1,-P4,P2,-P3]Lϕ(A1)Lϕ(F)Lϕ(M)V[Lϕ(F)Lϕ(A1)P4TLϕ(F)Lϕ(A1)P1TLϕ(F)Lϕ(A1)P3TLϕ(F)Lϕ(A1)P2T]+[P4,P1,P3,P2]Lϕ(A1)Lϕ(F)VT[Lϕ(M)Lϕ(F)Lϕ(A1)P1T-Lϕ(M)Lϕ(F)Lϕ(A1)P4TLϕ(M)Lϕ(F)Lϕ(A1)P2T-Lϕ(M)Lϕ(F)Lϕ(A1)P3T],
where
(2.19)P1=[In,0,0,0],P2=[0,In,0,0],P3=[0,0,In,0],P4=[0,0,0,In],ϕ(X0) is a particular symmetric solution to (2.13), and V is arbitrary real matrices with compatible sizes.Proof.
Suppose that (1.5) has a Hermitian solution X over ℍ. Applying properties (a) and (b) of ϕ(·) to (1.5) yields
(2.20)ϕ(A1)ϕ(X)=ϕ(C1),ϕ(X)ϕ(B2)=ϕ(C2),ϕ(A3)ϕ(X)ϕT(A3)=ϕ(C3),
implying that ϕ(X) is a real symmetric solution to (2.13).
Conversely, suppose that (2.13) has a real symmetric solution
(2.21)Ŷ=ŶT=(Yij)4×4,i,j=1,2,3,4.
That is,
(2.22)ϕ(A1)Ŷ=ϕ(C1),Ŷϕ(B2)=ϕ(C2),ϕ(A3)ŶϕT(A3)=ϕ(C3),
then by property (d) of ϕ(·),
(2.23)Tm-1ϕ(A1)TnŶ=Tm-1ϕ(C1)Tn,ŶTn-1ϕ(B2)Ts=Tn-1ϕ(C2)Ts,Tr-1ϕ(A3)TnŶTn-1ϕT(A3)Tr=Tr-1ϕ(C3)Tr,Rm-1ϕ(A1)RnŶ=Rm-1ϕ(C1)Rn,ŶRn-1ϕ(B2)Rs=Rn-1ϕ(C2)Rs,Rr-1ϕ(A3)RnŶRnϕT(A3)Rr=Rr-1ϕ(C3)Rr,Sm-1ϕ(A1)SnŶ=Sm-1ϕ(C1)Sn,ŶSn-1ϕ(B2)Ss=Sn-1ϕ(C2)Ss,Sr-1ϕ(A3)SnŶSn-1ϕT(A3)Sr=Sr-1ϕ(C3)Sr.
Hence,
(2.24)ϕ(A1)TnŶTn-1=ϕ(C1),TnŶTn-1ϕ(B2)=ϕ(C2),ϕ(A3)TnŶTn-1ϕT(A3)=ϕ(C3),ϕ(A1)RnŶRn-1=ϕ(C1),RnŶRn-1ϕ(B2)=ϕ(C2),ϕ(A3)RnŶRn-1ϕT(A3)=ϕ(C3),ϕ(A1)SnŶSn-1=ϕ(C1),SnŶSn-1ϕ(B2)=ϕ(C2),ϕ(A3)SnŶSn-1ϕT(A3)=ϕ(C3),
implying that TnŶTn-1,RnŶRn-1, and SnŶSn-1 are also symmetric solutions of (2.13). Thus,
(2.25)14(Ŷ+TnŶTn-1+RnŶRn-1+SnŶSn-1)
is a symmetric solution of (2.13), where
(2.26)Ŷ+TnŶTn-1+RnŶRn-1+SnŶSn-1=(Yij̃)4×4,i=1,2,3,4,Y11̃=Y11+Y22+Y33+Y44,Y12̃=Y12-Y12T+Y34-Y34T,Y13̃=Y13-Y13T+Y24T-Y24,Y14̃=Y14-Y14T+Y23-Y23T,Y21̃=Y12T-Y12+Y34T-Y34,Y22̃=Y11+Y22+Y33+Y44,Y23̃=Y14-Y14T+Y23-Y23T,Y24̃=Y13-Y13T+Y24T-Y24,Y31̃=Y13T-Y13+Y24-Y24T,Y32̃=Y14-Y14T+Y23-Y23T,Y33̃=Y11+Y22+Y33+Y44,Y34̃=Y12-Y12T+Y34-Y34T,Y41̃=Y14T-Y14+Y23T-Y23,Y42̃=Y13-Y13T+Y24T-Y24,Y43̃=Y12-Y12T+Y34-Y34T,Y44̃=Y11+Y22+Y33+Y44.
Let
(2.27)X̂=14(Y11+Y22+Y33+Y44)+14(Y12-Y12T+Y34-Y34T)i+14(Y13-Y13T+Y24T-Y24)j+14(Y14-Y14T+Y23-Y23T)k.
Then by (2.1),
(2.28)ϕ(X̂)=14(Ŷ+TnŶTn-1+RnŶRn-1+SnŶSn-1).
Hence, by the property (a), we know that X̂ is a Hermitian solution of (1.5). Observe that Yij,i,j=1,2,3,4, in (2.13) can be written as
(2.29)Yij=PiŶPjT.
From Lemma 2.2, the general Hermitian solution to (2.13) can be written as
(2.30)Ŷ=ϕ(X0)+4Lϕ(A1)Lϕ(F)Lϕ(M)VLϕ(A1)Lϕ(F)+4Lϕ(F)Lϕ(A1)VTLϕ(M)Lϕ(F)Lϕ(A1),
where V∈ℝ is arbitrary. Hence,
(2.31)Yij=Piϕ(X0)PjT+4PiLϕ(A1)Lϕ(F)Lϕ(M)VLϕ(A1)Lϕ(F)PjT+4PiLϕ(F)Lϕ(A1)VTLϕ(M)Lϕ(F)Lϕ(A1)PjT,
where i,j=1,2,3,4, substituting them into (2.14), yields the four real matrices X1,X2,X3, and X4 in (2.15)–(2.18).Now we consider the maximal and minimal ranks of four real matricesX1,X2,X3, and X4 in solution X=X1+X2i+X3j+X4k to (1.5) over ℍ.Theorem 2.7.
Suppose that system (1.5) over ℍ has a Hermitian solution, and A1=A11+A12i+A13j+A14k,C1=C11+C12i+C13j+C14k∈ℍm×n, B1=B11+B12i+B13j+B14k,C2=C21+C22i+C23j+C24k∈ℍn×s, A3=A31+A32i+A33j+A34k∈ℍr×n, C3=C31+C32i+C33j+C34k∈ℍr×r(2.32)S1={X1∈Rn×n∣A1X=C1,XB1=C2,A3XA3*=C3X=X1+X2i+X3j+X4k},S2={X2∈Rp×q∣A1X=C1,XB1=C2,A3XA3*=C3X=X1+X2i+X3j+X4k},S3={X3∈Rp×q∣A1X=C1,XB1=C2,A3XA3*=C3X=X1+X2i+X3j+X4k},S4={X4∈Rp×q∣A1X=C1,XB1=C2,A3XA3*=C3X=X1+X2i+X3j+X4k},L21=[C21C22C23C24],L11=[C11-C12-C13-C14],M31=[A32A33A34A31A34-A33-A34A31A32A33-A32A31],M11=[A12A13A14A11A14-A13-A14A11A12A13-A12A11],M12=[A11A13A14-A12A14-A13-A13A11A12-A14-A12A11],M13=[A11A12A14-A12A11-A13-A13-A14A12-A14A13A11],M14=[A11A12A13-A12A11A14-A13-A14A11-A14A13-A12],N11=[-B12B11B14-B13-B13-B14B11B12-B14B13-B12B11],N12=[B11B12B13B14-B13-B14B11B12-B14B13-B12B11],N13=[B11B12B13B14-B12B11B14-B13-B14B13-B12B11],N14=[B11B12B13B14-B12B11B14-B13-B13-B14B11B12].
Then the maximal and minimal ranks of Xi,i=1,2,3,4, in Hermitian solution X=X1+X2i+X3j+X4k to (1.5) are given by(2.33)max⁡Xi∈Sir(Xi)=min⁡{t1i,t},(2.34)min⁡Xi∈Sir(Xi)=2t1i+t-2t2i,
where
(2.35)t1i=r[L21N1iTL11M1i]-4r[B1*A1]+n,t=r[0M31TN11M11TM31ϕ(C3)ϕ(A3)ϕ(C2)ϕ(A3)ϕT(C1)N11TϕT(C2)ϕT(A3)ϕT(C2)ϕ(B1)ϕT(C2)ϕT(A1)M11ϕ(C1)ϕT(A3)ϕ(C1)ϕ(B1)ϕ(C1)ϕT(A1)]-8r[A3B1*A1]+2n,t2i=r[0N1iM1iTM31ϕ(A3)ϕ(C2)ϕ(A3)ϕT(C1)N11TϕT(C2)ϕ(B1)ϕT(C2)ϕT(A1)M11ϕ(C1)ϕ(B1)ϕ(C1)ϕT(A1)]-4r[A3B1*A1]-4r[B1*A1]+2n.Proof.
We only prove the case thati=1. Similarly, we can get the results that i=2,3,4. Let
(2.36)14P1ϕ(X0)P1T+14P2ϕ(X0)P2T+14P3ϕ(X0)P3T+14P4ϕ(X0)P4T=A,[P1,P2,P3,P4]Lϕ(A1)Lϕ(F)Lϕ(M)=B,[Lϕ(F)Lϕ(A1)P1TLϕ(F)Lϕ(A1)P2TLϕ(F)Lϕ(A1)P3TLϕ(F)Lϕ(A1)P4T]=C;
note that LM is Hermtian; then Lϕ(M) is symmetric; hence (2.15) can be written as
(2.37)X1=A+BVC+(BVC)*.
Note that A=A* and ℛ(B)⊆ℛ(C*); applying (2.9) and (2.10) to (2.37) yields
(2.38)max⁡X1∈S1r(X1)=min⁡{r[A,C*],r[ABB*0]},(2.39)min⁡X1∈S1r(X1)=2r[A,C*]+r[ABB*0]-2r[ABC0].
Let
(2.40)[P1,P2,P3,P4]=P,ai=[ϕ(Ai)0000ϕ(Ai)0000ϕ(Ai)0000ϕ(Ai)],i=1,3,b1=[ϕ(B1)0000ϕ(B1)0000ϕ(B1)0000ϕ(B1)].
Note that ϕ(X0) is a particular solution to (2.13), it is not difficult to find by Lemma 2.3, block Gaussian elimination, and property (e) of ϕ(·) that
(2.41)r[A,C*]=r[AP0b1T0a1]-4r[ϕ(A1)]-4r[ϕ(F)]=r[0P-14ϕT(C2)P1Tb1T-14ϕ(C1)P1Ta1]-4r[ϕ(B1*)ϕ(A1)]=r[0[P1,0,0,0]ϕT(C2)P1Tb1Tϕ(C1)P1Ta1]-4r[ϕ(B1*)ϕ(A1)]=r[L21N1iTL11M1i]-4r[ϕ(B1*)ϕ(A1)]+3r[ϕ(B1*)ϕ(A1)]+n=r[L21N1iTL11M1i]-4r[B1*A1]+n.
Note that LA=RA*, then Lϕ(A)=Rϕ*(A); hence(2.42)r[ABB*0]=r[AP000PT0a3Tb1a1T0a30000b1T0000a1000]-8r[ϕ(M)]-8r[ϕ(F)]-8r[ϕ(A1)]=r[0M31TN11M11TM31ϕ(C3)ϕ(A3)ϕ(C2)ϕ(A3)ϕT(C1)N11TϕT(C2)ϕT(A3)ϕT(C2)ϕ(B1)ϕT(C2)ϕT(A1)M11ϕ(C1)ϕT(A3)ϕ(C1)ϕ(B1)ϕ(C1)ϕT(A1)]-8r[ϕ(A3)ϕ(B1*)ϕ(A1)]+6r[ϕ(A3)ϕ(B1*)ϕ(A1)]+2n=r[0M31TN11M11TM31ϕ(C3)ϕ(A3)ϕ(C2)ϕ(A3)ϕT(C1)N11TϕT(C2)ϕT(A3)ϕT(C2)ϕ(B1)ϕT(C2)ϕT(A1)M11ϕ(C1)ϕT(A3)ϕ(C1)ϕ(B1)ϕ(C1)ϕT(A1)]-8r[A3B1*A1]+2n.
Similarly, we can obtain
(2.43)r[ABC0]=r[0N11M11TM31ϕ(A3)ϕ(C2)ϕ(A3)ϕT(C1)N11TϕT(C2)ϕ(B1)ϕT(C2)ϕT(A1)M11ϕ(C1)ϕ(B1)ϕ(C1)ϕT(A1)]-4r[A3B1*A1]-4r[B1*A1]+2n,
Substituting (2.41) and (2.43) into (2.38) and (2.39) yields (2.33) and (2.34), that is i=1.Corollary 2.8.
Suppose system (1.5) over ℍ have a Hermitian solution. Then we have the following.
(a) System (1.5) has a real hermtian solution if and only if
(2.44)2r[L21N1iTL11M1i]+r[0M31TN11M11TM31ϕ(C3)ϕ(A3)ϕ(C2)ϕ(A3)ϕT(C1)N11TϕT(C2)ϕT(A3)ϕT(C2)ϕ(B1)ϕT(C2)ϕT(A1)M11ϕ(C1)ϕT(A3)ϕ(C1)ϕ(B1)ϕ(C1)ϕT(A1)]=2r[0N1iM1iTM31ϕ(A3)ϕ(C2)ϕ(A3)ϕT(C1)N11TϕT(C2)ϕ(B1)ϕT(C2)ϕT(A1)M11ϕ(C1)ϕ(B1)ϕ(C1)ϕT(A1)]
hold when i=2,3,4. In that case, the real solution of (1.5) can be expressed as X=X1 in (2.15).
(b) System (1.5) has a complex solution if and only if (2.44) hold when i=3,4 or i=2,4 or i=2,3. In that case, the complex solutions of (1.5) can be expressed as X=X1+X2i or X=X1+X3j or X=X1+X4k, where X1,X2,X3, and X4 are expressed as (2.15), (2.16), (2.17), and (2.18), respectively.Proof.
From (2.34) we can get the necessary and sufficient conditions for Xi=0,i=1,2,3,4. Thus we can get the results of this Corollary.


## 3. Solvability Conditions for Real and Complex Hermitian Solutions to (1.6) Over ℍ

In this section, using the results of Theorem2.6, Theorem 2.7, and Corollary 2.8, we give necessary and sufficient conditions for (1.6) over ℍ to have real and complex Hermitian solutions.Theorem 3.1.
LetA1,A3,B1,C1,C2, and C3 be defined in Lemma 2.2, A4∈ℍl×n,C4∈ℍl×l, and suppose that system (1.5) and the matrix equation A4YA4*=C4 over ℍ have Hermitian solutions X and Y∈ℍn×n, respectively. Then system (1.6) over ℍ has a real Hermitian solution if and only if (2.44) hold when i=2,3,4, and
(3.1)r[0M31TM31ϕ(C3)]=2r(M31),(3.2)r[0M41T000M411Tϕ(B1)ϕT(A1)M41ϕ(C4)ϕ(A4)ϕ(C2)ϕ(A4)ϕT(C1)]=r(M41)+r[M41T00M411Tϕ(B1)ϕT(A1)],(3.3)r[00M41TM41M411ϕ(C4)0ϕT(B1)ϕT(C2)ϕT(A4)0ϕ(A1)ϕ(C1)ϕT(A4)]=r(M41)+r[M41M4110ϕT(B1)0ϕ(A1)],r[00M41T00000M411TϕT(A3)ϕ(B1)ϕT(A1)M41M411ϕ(C4)0000ϕ(A3)0ϕ(C3)ϕ(A3)ϕ(C2)ϕ(A3)ϕT(C1)0ϕT(B1)0ϕT(C2)ϕT(A3)ϕT(C2)ϕ(B1)ϕT(C2)ϕT(A1)0ϕ(A1)0ϕ(C1)ϕT(A3)ϕ(C1)ϕ(B1)ϕ(C1)ϕT(A1)]=2r[M41M4110ϕ(A3)0ϕT(B1)0ϕ(A1)],r[00M41T0000M411Tϕ(B1)ϕT(A1)M41M411ϕ(C4)000ϕT(B1)0ϕT(B1)ϕ(C2)ϕT(B1)ϕT(C1)0ϕ(A1)0ϕ(C1)ϕ(B1)ϕ(C1)ϕT(A1)]=2r[M41M4110ϕT(B1)0ϕ(A1)],
where
(3.4)M41=[A42A43A44A41A44-A43-A44A41A42A43-A42A41],M411=[A21000-A22000-A23000-A24000].Proof.
From Corollary2.8, system (1.5) over ℍ has a real Hermitian solution if and only if (2.44) hold when i=2,3,4. By (2.15), the real Hermitian solutions of (1.5) over ℍ can be expressed as
(3.5)X1=14P1ϕ(X0)P1T+14P2ϕ(X0)P2T+14P3ϕ(X0)P3T+14P4ϕ(X0)P4T+[P1,P2,P3,P4]Lϕ(A1)Lϕ(F)Lϕ(M)V[Lϕ(F)Lϕ(A1)P1TLϕ(F)Lϕ(A1)P2TLϕ(F)Lϕ(A1)P3TLϕ(F)Lϕ(A1)P4T]+[P1,P2,P3,P4]Lϕ(A1)Lϕ(F)VT[Lϕ(M)Lϕ(F)Lϕ(A1)P1TLϕ(M)Lϕ(F)Lϕ(A1)P2TLϕ(M)Lϕ(F)Lϕ(A1)P3TLϕ(M)Lϕ(F)Lϕ(A1)P4T],
where V is arbitrary matrices with compatible sizes.
LetA1,C1=0;B1,C2=0;A3=A4; C3=C4 in Corollary 2.8 and (2.15). It is easy to verify that the matrix equation A4YA4*=C4 over ℍ has a real Hermitian solution if and only if (3.1) hold and the real Hermitian solution can be expressed as
(3.6)Y1=14P1ϕ(Y0)P1T+14P2ϕ(Y0)P2T+14P3ϕ(Y0)P3T+14P4ϕ(Y0)P4T+[P1,P2,P3,P4]Lϕ(A4)U+UT[Lϕ(A1)P1TLϕ(A1)P2TLϕ(A1)P3TLϕ(A1)P4T],
where ϕ(Y0) is a particular solution to ϕ(A4)(Yij)4×4ϕT(A4)=ϕ(C4) and U is arbitrary matrices with compatible sizes. The expression of Y1 can also be obtained from Lemma 2.1. Let
(3.7)[P1,P2,P3,P4]=P,G=14P1ϕ(X0)P1T+14P2ϕ(X0)P2T+14P3ϕ(X0)P3T+14P4ϕ(X0)P4T-14P1ϕ(Y0)P1T-14P2ϕ(Y0)P2T-14P3ϕ(Y0)P3T-14P4ϕ(Y0)P4T.
Equating X1 and Y1, we obtain the following equation:
(3.8)X1-Y1=G+PLϕ(A1)Lϕ(F)Lϕ(M)V[Lϕ(F)Lϕ(A1)P1TLϕ(F)Lϕ(A1)P2TLϕ(F)Lϕ(A1)P3TLϕ(F)Lϕ(A1)P4T]+PLϕ(A1)Lϕ(F)VT[Lϕ(M)Lϕ(F)Lϕ(A1)P1TLϕ(M)Lϕ(F)Lϕ(A1)P2TLϕ(M)Lϕ(F)Lϕ(A1)P3TLϕ(M)Lϕ(F)Lϕ(A1)P4T]-PLϕ(A4)U-UT[Lϕ(A4)P1TLϕ(A4)P2TLϕ(A4)P3TLϕ(A4)P4T].
It is obvious that system (1.5) and the matrix equation A4YA4*=C4 over ℍ have common real Hermitian solution if and only if minr(X1-Y1)=0,thatis,X1-Y1=0. Hence, we have the matrix equation
(3.9)G=PLϕ(A4)U+UT[Lϕ(A4)P1TLϕ(A4)P2TLϕ(A4)P3TLϕ(A4)P4T]-PLϕ(A1)Lϕ(F)Lϕ(M)V[Lϕ(F)Lϕ(A1)P1TLϕ(F)Lϕ(A1)P2TLϕ(F)Lϕ(A1)P3TLϕ(F)Lϕ(A1)P4T]-PLϕ(A1)Lϕ(F)VT[Lϕ(M)Lϕ(F)Lϕ(A1)P1TLϕ(M)Lϕ(F)Lϕ(A1)P2TLϕ(M)Lϕ(F)Lϕ(A1)P3TLϕ(M)Lϕ(F)Lϕ(A1)P4T].
We know by Lemma 2.5 that (3.9) is solvable if and only if the following four rank equalities hold
(3.10)r[GPLϕ(A4)Rϕ(A4)PT0Rϕ(F)Rϕ(A1)PT0]=r[0PLϕ(A4)Rϕ(A4)PT0Rϕ(F)Rϕ(A1)PT0],r[GPLϕ(A4)PLϕ(A1)Lϕ(F)Rϕ(A4)PT00]=r[0PLϕ(A4)PLϕ(A1)Lϕ(F)Rϕ(A4)PT00],r[GPLϕ(A4)PLϕ(A1)Lϕ(F)Lϕ(M)Rϕ(A4)PT00Rϕ(M)Rϕ(F)Rϕ(A1)PT00]=r[0PLϕ(A4)PLϕ(A1)Lϕ(F)Lϕ(M)Rϕ(A4)PT00Rϕ(M)Rϕ(F)Rϕ(A1)PT00],r[GPLϕ(A4)PLϕ(A1)Lϕ(F)Rϕ(A4)PT00Rϕ(F)Rϕ(A1)PT00]=r[0PLϕ(A4)PLϕ(A1)Lϕ(F)Rϕ(A4)PT00Rϕ(F)Rϕ(A1)PT00].
Under the conditions that the system (1.5) and the matrix equation A4YA4*=C4 over ℍ have Hermitian solutions, it is not difficult to show by Lemma 2.3 and block Gaussian elimination that (3.10) are equivalent to the four rank equalities (3.2) and (3.3), respectively. Note that the processes are too much tedious; we omit them here. Obviously, the system (1.5) and the matrix equation A4YA4*=C4 over ℍ have a common real Hermitian solution if and only if (3.2) and (3.3) hold. Thus, the system (1.6) over ℍ has a real Hermitian solution if and only if (2.44) hold when i=2,3,4, and (3.1)–(3.3) hold.
Similarly, from Corollary2.8, we know that the system (1.5) over ℍ has a complex Hermitian solution if and only if (2.44) hold when i=3,4, i=2,4, or; i=2,3; its complex Hermitian solutions can be expressed as X=X1+X2i,X=X1+X3j, or X=X1+X4k. It is also easy to derive the necessary and sufficient condition for the matrix equation A4YA4*=C4 over ℍ to have a complex Hermitian solution; its complex Hermitian solution can be expressed as Y=Y1+Y2i,Y=Y1+Y3j,or Y=Y1+Y4k.By equating X1 and Y1,X2 and Y2,X3, and Y3,X4 and Y4, respectively, we can derive the necessary and sufficient conditions for the system (1.6) over ℍ to have a complex Hermitian solution.



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*Source: 307036-2012-01-23.xml*