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 equations A1X=C1,XB1=C2, and A3XA3*=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.
1. Introduction
Throughout this paper, we denote the real number field by ℝ; the complex field by ℂ; the set of all m×n matrices over the quaternion algebra
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
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
AXA*=B
over ℂ in terms of generalized inverses, respectively. In [10], Tian and Liu established the solvability conditions for
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
A1X=C1,XB2=C2,A3XA3*=C3,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,
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 Section 2, 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 matrix Mt=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
ϕ(Mt)=[Mt1Mt2Mt3Mt4-Mt2Mt1Mt4-Mt3-Mt3-Mt4Mt1Mt2-Mt4Mt3-Mt2Mt1].
By (2.1), it is easy to verify that ϕ(·) satisfies the following properties.
M=N⇔ϕ(M)=ϕ(N).
ϕ(kM+lN)=kϕ(M)+lϕ(N),ϕ(MN)=ϕ(M)ϕ(N),k,l∈ℝ.
ϕ(M*)=ϕT(M),
ϕ(M†)=ϕ†(M).
ϕ(M)=Tm-1ϕ(M)Tn=Rm-1ϕ(M)Rn=Sm-1ϕ(M)Sn, where t=m,n,
Tt=[0-It00It000000It00-It0],Rt=[00-It0000-ItIt0000It00],St=[000-It00It00-It00It000].
r[ϕ(M)]=4r(M).
M*=M⇔ϕT(M)=ϕ(M),
M*=-M⇔ϕT(M)=-ϕ(M).
The following lemmas provide us with some useful results over ℂ, which can be generalized to ℍ.
Lemma 2.1 (see [2, Lemma 2.1]).
Let A∈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
AA†B=B.
In that case, the general Hermitian solution of (1.3) can be expressed as
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]).
Let A1,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
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
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]).
Let A∈ℍm×n,B∈ℍm×k,C∈ℍl×n,D∈ℍj×k, and E∈ℍl×i. Then they satisfy the following rank equalities.
r(CLA)=r[AC]-r(A).
r[BALC]=r[BA0C]-r(C).
r[CRBA]=r[C0AB]-r(B).
r[ABLDREC0]=r[AB0C0E0D0]-r(D)-r(E).
Lemma 2.3 plays an important role in simplifying ranks of various block matrices.
Lemma 2.4 (see [11, Theorem 4.1, Corollary 4.2]).
Let A=±A*∈ℍm×m,B∈ℍm×n, andC∈ℍp×m be given; then
maxX∈Hn×pr[A-BXC∓(BXC)*]=min{r[ABC*],r[ABB*0],r[AC*C0]},minX∈Hn×pr[A-BXC∓(BXC)*]=2r[ABC*]+max{s1,s2},
where
s1=r[ABB*0]-2r[ABC*B*00],s2=r[AC*C0]-2r[ABC*C00].
If ℛ(B)⊆ℛ(C*),
maxX∈Hn×pr[A-BXC-(BXC)*]=min{r[AC*],r[ABB*0]},minX∈Hn×pr[A-BXC-(BXC)*]=2r[AC*]+r[ABB*0]-2r[ABC0].
Lemma 2.5 (see [28, Theorem 3.1]).
Let A∈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
B1X1+X2C2+B3X3C3+B4X4C4=A
is consistent if and only if
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
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
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
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],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],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],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
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
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
Ŷ=ŶT=(Yij)4×4,i,j=1,2,3,4.
That is,
ϕ(A1)Ŷ=ϕ(C1),Ŷϕ(B2)=ϕ(C2),ϕ(A3)ŶϕT(A3)=ϕ(C3),
then by property (d) of ϕ(·),
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,
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,
14(Ŷ+TnŶTn-1+RnŶRn-1+SnŶSn-1)
is a symmetric solution of (2.13), where
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
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),
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
Yij=PiŶPjT.
From Lemma 2.2, the general Hermitian solution to (2.13) can be written as
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,
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 matrices X1,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×rS1={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 bymaxXi∈Sir(Xi)=min{t1i,t},minXi∈Sir(Xi)=2t1i+t-2t2i,
where
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 that i=1. Similarly, we can get the results that i=2,3,4. Let
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
X1=A+BVC+(BVC)*.
Note that A=A* and ℛ(B)⊆ℛ(C*); applying (2.9) and (2.10) to (2.37) yields
maxX1∈S1r(X1)=min{r[A,C*],r[ABB*0]},minX1∈S1r(X1)=2r[A,C*]+r[ABB*0]-2r[ABC0].
Let
[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
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); hencer[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
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
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 Theorem 2.6, 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.
Let A1,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
r[0M31TM31ϕ(C3)]=2r(M31),r[0M41T000M411Tϕ(B1)ϕT(A1)M41ϕ(C4)ϕ(A4)ϕ(C2)ϕ(A4)ϕT(C1)]=r(M41)+r[M41T00M411Tϕ(B1)ϕT(A1)],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
M41=[A42A43A44A41A44-A43-A44A41A42A43-A42A41],M411=[A21000-A22000-A23000-A24000].
Proof.
From Corollary 2.8, 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
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.
Let A1,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
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
[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:
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
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
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 Corollary 2.8, 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.
Acknowledgment
This research was supported by the Excellent Young Teacher Grant of East China University of Science and Technology (yk0157124) and the Natural Science Foundation of China (11001079).
HungerfordT. W.198073New York, NY, USASpringerxxiii+502Graduate Texts in Mathematics60065410.1007/978-1-4612-6101-8ZBL0442.00002KhatriC. G.MitraS. K.Hermitian and nonnegative definite solutions of linear matrix equations1976314579585041721210.1137/0131050ZBL0359.65033HuaD.LancasterP.Linear matrix equations from an inverse problem of vibration theory19962463147140765710.1016/0024-3795(94)00311-4ZBL0861.15014LiuY. H.TianY. G.More on extremal ranks of the matrix expressions A−BX±X∗B∗ with statistical applications2008154307325241377510.1002/nla.553ZBL1212.15029WeiM.WangQ.On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation AXA∗=B2007846945952233537710.1080/00207160701458344ZBL1129.15012ZhangX.ChengM.-Y.The rank-constrained Hermitian nonnegative-definite and positive-definite solutions to the matrix equation AXA∗=B2003370163174199432510.1016/S0024-3795(03)00385-9ZBL1026.15011LiuY. H.TianY. G.Max-min problems on the ranks and inertias of the matrix expressions A−BXC±(BXC)∗ with applications20111483593622276988110.1007/s10957-010-9760-8ZBL1223.90077GroßJ.A note on the general Hermitian solution to AXA∗=B199821257621698241ZBL1006.15011LiuY. H.TianY. G.TakaneY.Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA∗=B20094311223592372256302810.1016/j.laa.2009.03.011ZBL1180.15018TianY. G.LiuY. H.Extremal ranks of some symmetric matrix expressions with applications200628389090510.1137/S08954798024155452262987ZBL1123.15001TianY. G.Maximization and minimization of the rank and inertia of the Hermitian matrix expression A−BX−(BX)∗
with applications20114341021092139278168110.1016/j.laa.2010.12.010ZBL1211.15022WangQ. W.WuZ.-C.Common Hermitian solutions to some operator equations on Hilbert C∗
-modules20104321231593171263927610.1016/j.laa.2010.01.015TianY. G.Ranks of solutions of the matrix equation AXB=C2003512111125197685710.1080/0308108031000114631LiuY. H.Ranks of solutions of the linear matrix equation AX+YB=C2006526-7861872228155310.1016/j.camwa.2006.05.011ZBL1129.15009LiuY. H.Ranks of least squares solutions of the matrix equation AXB=C200855612701278239436610.1016/j.camwa.2007.06.023ZBL1157.15014TianY. G.Upper and lower bounds for ranks of matrix expressions using generalized inverses2002355187214193014510.1016/S0024-3795(02)00345-2ZBL1016.15003TianY. G.ChengS.The maximal and minimal ranks of A−BXC with applications200393453622028174ZBL1036.15004TianY. G.The maximal and minimal ranks of some expressions of generalized inverses of matrices2002254745755193467110.1007/s100120200015ZBL1007.15005TianY. G.The minimal rank of the matrix expression A−BX−YC200214140481883608ZBL1032.15001WangQ. W.WuZ.-C.LinC.-Y.Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications2006182217551764228261710.1016/j.amc.2006.06.012ZBL1108.15014WangQ. W.SongG.-J.LinC.-Y.Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application2007189215171532233210710.1016/j.amc.2006.12.039ZBL1124.15010WangQ. W.YuS.-W.LinC.-Y.Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications20081952733744238125210.1016/j.amc.2007.05.018ZBL1149.15012YuS. W.SongG. J.Extreme ranks of Hermitian solution to a pair of matrix equationsProceedings of the 6th International Workshop on Matrix and OperatorsJuly 2011Chengdu, China253255WangQ. W.JiangJ.Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation2010205525732735973ZBL1207.15016WangQ. W.YuS. W.XieW.Extreme ranks of real matrices in solution of the quaternion matrix equation AXB=C with applications20101723453602654356ZBL1188.15016WangQ. W.YuS.-W.ZhangQ.The real solutions to a system of quaternion matrix equations with applications200937620602079253076310.1080/00927870802317590WangQ. W.ZhangH.-S.YuS.-W.On solutions to the quaternion matrix equation AXB+CYD=E2008173433582430860ZBL1154.15019TianY. G.The solvability of two linear matrix equations2000482123147181344010.1080/03081080008818664ZBL0970.15005