# A Real Representation Method for Solving Yakubovich-j-Conjugate Quaternion Matrix Equation

Authors: Caiqin Song; Jun-e Feng; Xiaodong Wang; Jianli Zhao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A new approach is presented for obtaining the solutions to Yakubovich-j-conjugate quaternion
matrix equation X−AX^B=CY based on the real representation of a quaternion matrix. Compared
to the existing results, there are no requirements on the coefficient matrix A. The closed
form solution is established and the equivalent form of solution is given for this Yakubovich-j-conjugate
quaternion matrix equation. Moreover, the existence of solution to complex conjugate
matrix equation X−AX¯B=CY is also characterized and the solution is derived in an explicit
form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix
equation over complex field is a special case of Yakubovich-j-conjugate quaternion matrix equation
X−AX^B=CY. Numerical example shows the effectiveness of the proposed results.


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## Body

## 1. Introduction

The linear matrix equationX-AXB=C, which is called the Kalman-Yakubovich matrix equation in [1], is closely related to many problems in conventional linear control systems theory, such as pole assignment design [2], Luenberger-type observer design [3, 4], and robust fault detection [5, 6]. In recent years, many studies have been reported on the solutions to many algebraic equations including quaternion matrix equations and nonlinear matrix equations. Yuan and Liao [7] investigated the least squares solution of the quaternion j-conjugate matrix equation X-AX^B=C (where X^ denotes the j-conjugate of quaternion matrix X) with the least norm using the complex representation of quaternion matrix, the Kronecker product of matrices, and the Moore-Penrose generalized inverse. The authors in [8] considered the matrix nearness problem associated with the quaternion matrix equation AXAH+BYBH=C by means of the CCD-Q, GSVD-Q, and the projection theorem in the finite dimensional inner product space. In addition, Song et al. [9, 10] established the explicit solutions to the quaternion j-conjugate matrix equation X-AX^B=C,  XF-AX^=CY, but here the known quaternion matrix A is a block diagonal form. Wang et al. in [11, 12] investigated Hermitian tridiagonal solutions and the minimal-norm solution with the least norm of quaternionic least squares problem in quaternionic quantum theory. Besides, in [13, 14], some solutions for the Kalman-Yakubovich equation are presented in terms of the coefficients of characteristic polynomial of matrix A or the Leverrier algorithm. The existence of solution to the matrix equation X-AX¯B=C, which, for convenience, is called the Kalman-Yakubovich-conjugate matrix equation, is established, and the explicit solution is derived. Several necessary and sufficient conditions for the existence of a unique solution to the matrix equation ∑i=0k‍AiXBi=E over quaternion field are obtained [15]. The authors in [16–18] have provided the consistence of the matrix equation AX-X¯B=C via the consimilarity of two matrices. In [19], Wu et al. construct some explicit expressions of the solution of the matrix equation AX-X¯B=C by means of a real representation of a complex matrix. It is shown that there exists a unique solution if and only if AA¯ and BB¯ have no common eigenvalues.In this paper, we study quaternionj-conjugate matrix equation X-AX^B=CY by means of real representation of a quaternion matrix. Compared to the complex representation method [9, 10], the real representation method does not require any special case of the known matrix A. We propose the explicit solutions to the above Yakubovich-j-conjugate quaternion matrix equation. As the special case of quaternion j-conjugate matrix equation X-AX^B=CY, complex conjugate matrix equation X-AX¯B=C and Kalman-Yakubovich quaternion matrix equation are also investigated. The explicit solutions to the complex conjugate matrix equation have been established.Throughout this paper, we use the following notations. LetR denote the real number field, C the complex number field, and Q=R⊕Ri⊕Rj⊕Rk the quaternion field, where i2=j2=k2=-1,  ij=-ji=k. Rm×n(Cm×n or Qm×n) denotes the set of all m×n matrices on R(CorQ). For any matrix A∈Cm×n, AT, A¯, AH, detA, and A* represent the transpose, conjugate, conjugate transpose, determinant, and adjoint of A, respectively. In addition, symbol Aσ is the real representation of quaternion matrix A. A⊗B=(aijB) denotes the Kronecker product of two matrices A and B. If A∈Qm×n, let A=A1+A2i+A3j+A4k, where At∈Rm×n,  t=1,…,4, and define A^=A1-A2i+A3j-A4k to be the j-conjugate of A. For A∈Cm×n, vec(A) is defined as vec(A)=[a1Ta2T⋯anT]T. Furthermore, letting A∈Qn×n, B∈Qn×r, and C∈Qm×n, we have the following notations associated with these matrices:
(1)Qc(A,B,n)=[BAB⋯An-1B],Qo(A,C,k)=[CCA⋮CAk-1],fAσ(s)=det(sI-Aσ)=s2n+α2n-1s2n-1+⋯+α1s+α0,Sr(I,Aσ)=[Irα2Irα4Ir⋯α2(n-1)IrIrα2Ir⋯α2(n-2)Ir⋯⋯Irα2IrIr].
Obviously, Qc(A,B,n) is the controllability matrix of the matrix pair (A,B), Qo(A,C,k) is the observability matrix of the matrix pair (A,C), and Sr(I,Aσ) is a symmetric matrix.


## 2. Quaternion-j-Conjugate Matrix Equation X-AX^B=CY

### 2.1. Real Matrix EquationX-AXB=CY

In this subsection, we investigate the Yakubovich matrix equation over real field(2)X-AXB=CY.Theorem 1.
Suppose the real matricesA∈Rn×n, B∈Rp×p, C∈Rn×r, {s∣det(I-sA)=0}∩λ(B)=ϕ; let
(3)f(I,A)(s)=det(I-sA)=αnsn+⋯+α1s+α0,α0=1,adj(I-sA)=Rn-1sn-1+⋯+R1s+R0.
Then, all the solutions to the Yakubovich matrix equation (2) can be established as
(4)X=∑i=0n-1RiCZBi,Y=Zf(I,A)(B),
where the matrix Z∈Rr×p is an arbitrary matrix.Proof.
We first show that the matricesX and Y given in (4) are solutions of the matrix equation (2). By the direct calculation we have
(5)X-AXB=∑i=0n-1RiCZBi-A∑i=0n-1RiCZBiB=∑i=0n-1RiCZBi-∑i=0n-1ARiCZBi+1=R0CZ+∑i=1n-1(Ri-ARi-1)CZBi-ARn-1CZBn.
Due to the relation (I-sA)adj(I-sA)=Idet(I-sA), it is easily derived that
(6)R0=α0I,Ri-ARi-1=αiI,i=1:n-1,-ARn-1=αnI.
So one has
(7)R0CZ+∑i=1n-1(Ri-ARi-1)CZBi-ARn-1CZBn=CZ∑i=0nαiBi=CZf(I,A)(B)=CY.
Thus, the matrices X and Y given in (4) satisfy the matrix equation (2).
Secondly, we show the completeness of solution (4). It follows from Theorem 6 of [20] that there are rp degrees of freedom in the solution of matrix equation (2), while solution (4) has exactly rp parameters represented by the elements of the free matrix Z. Therefore, in the following we only need to show that all the parameters in the matrix Z contribute to the solution. To do this, it suffices to show that the mapping Z→(X,Y) defined by (5) is injective. This is true since f(I,A)(B) is nonsingular under the condition of {s∣det(I-sA)=0}∩λ(B)=ϕ. The proof is thus completed.In [21], we can find the following well-known generalized Faddeev-Leverrier algorithm:
(8)Rk=Rk-1A+αkIn,R0=In,k=1,2,…,n,αk=trace(Rk-1A)k,α0=1,k=1,2,…,n,
where αi, i=0,1,2,…,n-1, are the coefficients of the characteristic polynomial of the matrix A, and Ri, i=0,1,…,n-1, are the coefficient matrices of the adjoint matrix adj(sIn-A).Theorem 2.
Given matricesA∈Rn×n, B∈Rp×p, C∈Rr×p, let
(9)f(I,A)(s)=det(I-sA)=αnsn+⋯+α1s+α0,α0=1.
Then the matrices X and Y given by (4) have the following equivalent form:
(10)X=∑j=0n-1∑k=0jαkAj-kCZBj,Y=Zf(I,A)(B).Proof.
According to (8), the following is easily obtained:
(11)R0=I,R1=α1I+A,R2=α2I+α1A+A2,⋮Rn-1=αn-1I+αn-2A+⋯+An-1.
This relation can be compactly expressed as
(12)Rj=∑k=0jαkAj-k,α0=1,j=1,2,…,n-1.
Substituting this into the expression of X in (10) and recording the sum, we have
(13)X=∑j=0n-1RjCZBj=∑j=0n-1(∑k=0jαkAj-k)CZBj=∑j=0n-1∑k=0jαkAj-kCZBj.
Combining this with Theorem 1 gives the conclusion.


### 2.2. Real Representation of a Quaternion Matrix

For any quaternion matrixA=A1+A2i+A3j+A4k∈Qm×n, Al∈Rm×n  (l=1,2,3,4), the real representation matrix of quaternion matrix A can be defined as
(14)Aσ=[A1A2-A3A4A2-A1-A4-A3A3-A4A1A2A4A3A2-A1]∈R4m×4n.For am×n quaternion matrix A, we define Aσt=(Aσ)t. In addition, if we let
(15)Pt=[It0000-It0000It0000-It],Qt=[0-It00It000000It00-It0],St=[000-It00It00-It00It000],Rt=[00It0000It-It0000-It00],
in which It is a t×t identity matrix, then Pt, Qt, St, Rt are unitary matrices.The real representation has the following properties, which are given in [13].Proposition 3.
LetA,B∈Qm×n, C∈Qn×s, a∈R. Then(1)
(A+B)σ=Aσ+Bσ, (aA)σ=aAσ, (AC)σ=AσPnCσ=Aσ(C^)σPs;(2)
A=B⇔Aσ=Bσ;(3)
Qm-1AσQn=-Aσ, Rm-1AσRn=Aσ, Sm-1AσSn=-Aσ, Pm-1AσPn=(A^)σ;(4)
the quaternion matrixAis nonsingular if and only ifAσis nonsingular, and the quaternion matrixAis an unitary matrix if and only ifAσis an orthogonal matrix;(5)
ifA∈Qm×m, thenAσ2k=((AA^)k)σPm;(6)
A∈Qm×m,B∈Qn×n,C∈Qm×n,
andk+lis even, then(16)AσkCσBσl={((AA^)s(AC^B)(B^B)t)σ,k=2s+1,l=2t+1,((AA^)sC(B^B)t)σ,k=2s,l=2t.Proposition 4.
Ifλ is a characteristic value of Aσ, then so are ±λ, ±λ¯.For anyA∈Qm×m, let the characteristic polynomial of the real representation matrix Aσ be f(I,Aσ)(λ)=det(I4m-λAσ)=∑k=02m‍a2kλ2k, and define hAσ(λ)=λ4mf(I,Aσ)(λ-1)=∑k=02m‍a2kλ2(2m-k). So by Propositions 3 and 4 we have the following proposition.Proposition 5.
LetA∈Qm×m, B∈Qn×n. Then(1)
f(I,Aσ)(λ)is a real polynomial, andf(I,Aσ)(λ)=∑k=02m‍a2kλ2k;(2)
hAσ(λ)is a real polynomial, andhAσ(λ)=∑k=02m‍a2kλ2(2m-k);(3)
hAσ(Bσ)=(gAσ(BB~))σPn,f(I,Aσ)(Bσ)=(pAσ(BB~))σPn,in whichgAσ(λ)=∑k=02m‍a2kλm-k, pAσ(λ)=∑k=02m‍a2kλkare real polynomials.Proof.
By Proposition4, we easily know that ak is a real number, and a2k+1=0. For any k, by Proposition 3, we have Bσ2k=((BB~)k)σPn, so we can obtain the result (3).


### 2.3. On Solutions to the Quaternionj-Conjugate Matrix Equation X-AX^B=CY

In this subsection, we discuss the solution of the following quaternion matrix equation:(17)X-AX^B=CY,
by means of real representation, where A∈Qn×n, B∈Qp×p, and C∈Qn×r are known matrices, X∈Qn×p and Y∈Qr×p are unknown matrices.We first define the real representation of quaternion matrix equation (17) by
(18)V-AσVBσ=CσPrW.According to (1) in Proposition3, the quaternion matrix equation (17) is equivalent to the following equation:
(19)(X-AX^B)σ=Xσ-AσXσBσ.
Therefore, the matrix equation (17) can be converted into
(20)Xσ-AσXσBσ=CσPrYσ.
Thus, we have the following conclusion.Proposition 6.
Given the quaternion matricesA∈Qn×n, B∈Qp×p and C∈Qn×r, then the quaternion matrix equation (17) has a solution (X,Y) if and only if the real representation matrix equation (18) has a solution (V,W)=(Xσ,Yσ).Theorem 7.
LetA∈Qn×n, B∈Qp×p, and C∈Qn×r. Then quaternion matrix equation (17) has a solution (X,Y) if and only if real representation matrix equation (18) has a solution (V,W). Furthermore, if (V,W) is a solution to (18), then the following quaternion matrices are solutions to quaternion matrix equation (17):
(21)X=116[IniInjInkIn]×(V-Qn-1VQp+Rn-1VRp-Sn-1VSp)[Ip-iIp-jIp-kIp],Y=116[IriIrjIrkIr]×(W-Qn-1WQp+Rn-1WRp-Sn-1WSp)[Ip-iIp-jIp-kIp].Proof.
By (3) of Proposition3, the quaternion matrix equation (18) is equivalent to
(22)V-Rn-1AσRnVRp-1BσRp=Rn-1CσRrPrW.
After multiplying the two sides of quaternion matrix equation (22) by Rp-1, we can obtain
(23)VRp-1-Rn-1AσRnVRp-1Bσ=Rn-1CσRrPrWRp-1.
Before multiplying the two sides of quaternion matrix equation (23) by Rn, we have
(24)RnVRp-1-AσRnVRp-1Bσ=CσRrPrWRp-1.
Noting that  Rp-1=-Rp, RrPr=PrRr, we give
(25)Rn-1VRp-AσRn-1VRpBσ=CσPrRr-1WRp.
This shows that if (V,W) is a real solution of matrix equation (18), then (Rn-1VRp,Rr-1WRp) is also a real solution of quaternion matrix equation (18). In addition, according to (3) of Proposition 3, the quaternion matrix equation (18) is also equivalent to
(26)V-QnAσQnVQpBσQp=QnCσQrPrW.
After multiplying the two sides of quaternion matrix equation (26) by Qp-1, we have
(27)VQp-1-QnAσQnVQpBσ=QnCσQrPrWQp-1.
Noting that  Qp-1=-Qp, QrPr=-PrQr, before multiplying the two sides of the quaternion matrix equation (27) by Qn-1, gives
(28)(-Qn-1VQp)-Aσ(-Qn-1VQp)Bσ=CσPp(-Qr-1WQp).
This is to say that if (V,W) is a real solution of matrix equation (18), then (-Qn-1VQp,-Qr-1WQp) is also a real solution of matrix equation (18). Similarly, we can prove that (-Sn-1VSp,-Sr-1WSp) is also a real solution of quaternion matrix equation (18). In this case, the conclusion can be obtained along the line of the proof of Theorem  4.2 in [13].Theorem 8.
Given the quaternion matricesA∈Qn×n, B∈Qp×p, and C∈Qn×r, let
(29)f(I,Aσ)(s)=det(I4n-sAσ)=∑k=02na2ks2k,pAσ(s)=∑k=02na2ksk.
Then the matrices X∈Qn×p, Y∈Qr×p are given by
(30)X=∑k=02n-1∑s=k2n-1α2k(AA^)s-kCZ(B^B)s+∑k=02n-1∑s=k2n-1α2k(AA^)s-kAC^Z^B(B^B)s,Y=ZpAσ(B^B),
in which Z is an arbitrary quaternion matrix.Proof.
If Yakubovich quaternionj-conjugate matrix equation (17) has solution (X,Y), then real representation matrix equation (18) has solution (V,W)=(Xσ,Yσ) with the free parameter Zσ. By Theorems 2 and 7, we have
(31)Xσ=∑k=02n-1∑j=0kαjAσj-kCσPrZσBσj=∑k=02n-1∑j=2k4n-1α2kAσj-2kCσPrZσBσj=∑k=02n-1α2k[∑s=k2n-1Aσ2s-2kCσPrZσBσ2s+∑s=k2n-1‍Aσ2s-2k+1CσPrZσBσ2s+1]=∑k=02n-1α2k×[∑s=k2n-1((AA^)s-k)σPnCσPrZσ((B^B)s)σPp+∑s=k2n-1((AA^)s-k)σPnAσCσPrZσBσ((B^B)s)σPp]=∑k=02n-1α2k[∑s=k2n-1((AA^)s-kCZ(B^B)s)σ+∑s=k2n-1((AA^)s-kAC^Z^B(B^B)s)σ].
In addition, by Proposition 5, f(I,Aσ)(s) is a real polynomial and f(I,Aσ)(Bσ)=(pAσ(BB^))σPp. So according to Proposition 3, we obtain
(32)Yσ=Zσf(I,Aσ)(Bσ)=Zσ(pAσ(BB^))σPp=(ZpAσ(B^B))σ.
Thus, the conclusion above has been proved.In the following, we provide an equivalent statement of Theorem8.Theorem 9.
Given quaternion matricesA∈Qn×n, B∈Qp×p, and C∈Qn×r, let
(33)f(I,Aσ)(s)=det(I4n-sAσ)=∑k=02na2ks2k,pAσ(s)=∑k=02na2ksk.
Then the matrices X∈Qn×p, Y∈Qr×p given by (30) have the following equivalent form:
(34)X=Qc(AA^,C,2n)Sr(I,Aσ)Qo(BB^,Z,2n)+Qc(AA^,AC^,2n)Sr(I,Aσ)Qo(B^B,Z^B,2n),Y=ZpAσ(B^B),
in which Z is an arbitrary quaternion matrix.Proof.
By the direct computation, we have(35)∑k=02n-1∑s=k2n-1α2k(AA^)s-kCZ(B^B)s=Qc(AA^,C,n)Sr(I,Aσ)Qo(BB^,Z,2n),∑k=02n-1∑s=k2n-1α2k(AA^)s-kAC^Z^B(B^B)s=Qc(AA^,AC^,2n)Sr(I,Aσ)Qo(B^B,Z^B,2n).
Thus, the first conclusion has been proved. With this the second conclusion is obviously true.Finally, we consider the solution to the so-called Kalman-Yakubovichj-conjugate quaternion matrix equation
(36)X-AX^B=C.
Based on the main result proposed above, we have the following conclusions regarding the matrix equation (36).Corollary 10.
Given quaternion matricesA∈Qn×n, B∈Qp×p, and C∈Qn×p, let
(37)f(I,Aσ)(s)=det(I4n-sAσ)=∑k=02na2ks2k,pAσ(s)=∑k=02na2ksk.
If X is a solution of equation (36), then
(38)XpAσ(B^B)=∑k=02n-1∑j=k2n-1α2k(AA^)j-kC(B^B)j+∑k=02n-1∑j=k2n-1(AA^)j-kAC^B(B^B)j.Proof.
IfX is a solution of equation (36), then Y=Xσ is a solution of the equation Xσ-AσXσBσ=Cσ. By Theorem  3 in [22] and Proposition 3, we have
(39)Xσf(I,Aσ)(Bσ)=∑k=02n-1∑j=2k4n-1α2kAσj-2kCσBσj.
By Proposition 5, f(I,Aσ)(s) is a real polynomial and f(I,Aσ)(Bσ)=(pAσ(BB^))σPp. So from Proposition 3 and (39), we have
(40)[XpAσ(BB^)]σ=Xσ[pAσ(B^B)]σPp=Xσf(I,Aσ)(Bσ)=∑k=02n-1∑j=2k4n-1α2kAσj-2kCσBσj=∑k=02n-1α2k[∑j=k2n-1Aσ2j-2kCσBσ2j+∑j=k2n-1Aσ2j+1-2kCσBσ2j+1]=∑k=02n-1α2k[∑j=k2n-1((AA^)j-k)σPnCσ((B^B)j)σPp+∑j=k2n-1((AA^)j-k)σPnAσCσBσ((B^B)j)σPp]=∑k=02n-1α2k×[∑j=k2n-1((AA^)j-kC(B^B)j)σ+∑j=k2n-1((AA^)j-kAC^B(B^B)j)σ]=∑k=02n-1∑j=k2n-1α2k((AA^)j-kC(B^B)j)σ+∑k=02n-1∑j=k2n-1((AA^)j-kAC^B(B^B)j)σ.
Thus, the first conclusion has been proved. With this the second conclusion is obviously true.In the following, we provide an equivalent statement of Theorem7.Corollary 11.
Given quaternion matricesA∈Qn×n, B∈Qp×p, and C∈Qn×p, let
(41)f(I,Aσ)(s)=det(I4n-sAσ)=∑k=02na2ks2k,pAσ(s)=∑k=02na2ksk.
If X is a solution of (36), then
(42)XpAσ(B^B)=Qc(AA^,C,2n)Sp(I,Aσ)Qo(B^B,Ip,2n)+Qc(AA^,A,2n)Sn(I,Aσ)Qo(B^B,C^B,2n).


## 2.1. Real Matrix EquationX-AXB=CY

In this subsection, we investigate the Yakubovich matrix equation over real field(2)X-AXB=CY.Theorem 1.
Suppose the real matricesA∈Rn×n, B∈Rp×p, C∈Rn×r, {s∣det(I-sA)=0}∩λ(B)=ϕ; let
(3)f(I,A)(s)=det(I-sA)=αnsn+⋯+α1s+α0,α0=1,adj(I-sA)=Rn-1sn-1+⋯+R1s+R0.
Then, all the solutions to the Yakubovich matrix equation (2) can be established as
(4)X=∑i=0n-1RiCZBi,Y=Zf(I,A)(B),
where the matrix Z∈Rr×p is an arbitrary matrix.Proof.
We first show that the matricesX and Y given in (4) are solutions of the matrix equation (2). By the direct calculation we have
(5)X-AXB=∑i=0n-1RiCZBi-A∑i=0n-1RiCZBiB=∑i=0n-1RiCZBi-∑i=0n-1ARiCZBi+1=R0CZ+∑i=1n-1(Ri-ARi-1)CZBi-ARn-1CZBn.
Due to the relation (I-sA)adj(I-sA)=Idet(I-sA), it is easily derived that
(6)R0=α0I,Ri-ARi-1=αiI,i=1:n-1,-ARn-1=αnI.
So one has
(7)R0CZ+∑i=1n-1(Ri-ARi-1)CZBi-ARn-1CZBn=CZ∑i=0nαiBi=CZf(I,A)(B)=CY.
Thus, the matrices X and Y given in (4) satisfy the matrix equation (2).
Secondly, we show the completeness of solution (4). It follows from Theorem 6 of [20] that there are rp degrees of freedom in the solution of matrix equation (2), while solution (4) has exactly rp parameters represented by the elements of the free matrix Z. Therefore, in the following we only need to show that all the parameters in the matrix Z contribute to the solution. To do this, it suffices to show that the mapping Z→(X,Y) defined by (5) is injective. This is true since f(I,A)(B) is nonsingular under the condition of {s∣det(I-sA)=0}∩λ(B)=ϕ. The proof is thus completed.In [21], we can find the following well-known generalized Faddeev-Leverrier algorithm:
(8)Rk=Rk-1A+αkIn,R0=In,k=1,2,…,n,αk=trace(Rk-1A)k,α0=1,k=1,2,…,n,
where αi, i=0,1,2,…,n-1, are the coefficients of the characteristic polynomial of the matrix A, and Ri, i=0,1,…,n-1, are the coefficient matrices of the adjoint matrix adj(sIn-A).Theorem 2.
Given matricesA∈Rn×n, B∈Rp×p, C∈Rr×p, let
(9)f(I,A)(s)=det(I-sA)=αnsn+⋯+α1s+α0,α0=1.
Then the matrices X and Y given by (4) have the following equivalent form:
(10)X=∑j=0n-1∑k=0jαkAj-kCZBj,Y=Zf(I,A)(B).Proof.
According to (8), the following is easily obtained:
(11)R0=I,R1=α1I+A,R2=α2I+α1A+A2,⋮Rn-1=αn-1I+αn-2A+⋯+An-1.
This relation can be compactly expressed as
(12)Rj=∑k=0jαkAj-k,α0=1,j=1,2,…,n-1.
Substituting this into the expression of X in (10) and recording the sum, we have
(13)X=∑j=0n-1RjCZBj=∑j=0n-1(∑k=0jαkAj-k)CZBj=∑j=0n-1∑k=0jαkAj-kCZBj.
Combining this with Theorem 1 gives the conclusion.


## 2.2. Real Representation of a Quaternion Matrix

For any quaternion matrixA=A1+A2i+A3j+A4k∈Qm×n, Al∈Rm×n  (l=1,2,3,4), the real representation matrix of quaternion matrix A can be defined as
(14)Aσ=[A1A2-A3A4A2-A1-A4-A3A3-A4A1A2A4A3A2-A1]∈R4m×4n.For am×n quaternion matrix A, we define Aσt=(Aσ)t. In addition, if we let
(15)Pt=[It0000-It0000It0000-It],Qt=[0-It00It000000It00-It0],St=[000-It00It00-It00It000],Rt=[00It0000It-It0000-It00],
in which It is a t×t identity matrix, then Pt, Qt, St, Rt are unitary matrices.The real representation has the following properties, which are given in [13].Proposition 3.
LetA,B∈Qm×n, C∈Qn×s, a∈R. Then(1)
(A+B)σ=Aσ+Bσ, (aA)σ=aAσ, (AC)σ=AσPnCσ=Aσ(C^)σPs;(2)
A=B⇔Aσ=Bσ;(3)
Qm-1AσQn=-Aσ, Rm-1AσRn=Aσ, Sm-1AσSn=-Aσ, Pm-1AσPn=(A^)σ;(4)
the quaternion matrixAis nonsingular if and only ifAσis nonsingular, and the quaternion matrixAis an unitary matrix if and only ifAσis an orthogonal matrix;(5)
ifA∈Qm×m, thenAσ2k=((AA^)k)σPm;(6)
A∈Qm×m,B∈Qn×n,C∈Qm×n,
andk+lis even, then(16)AσkCσBσl={((AA^)s(AC^B)(B^B)t)σ,k=2s+1,l=2t+1,((AA^)sC(B^B)t)σ,k=2s,l=2t.Proposition 4.
Ifλ is a characteristic value of Aσ, then so are ±λ, ±λ¯.For anyA∈Qm×m, let the characteristic polynomial of the real representation matrix Aσ be f(I,Aσ)(λ)=det(I4m-λAσ)=∑k=02m‍a2kλ2k, and define hAσ(λ)=λ4mf(I,Aσ)(λ-1)=∑k=02m‍a2kλ2(2m-k). So by Propositions 3 and 4 we have the following proposition.Proposition 5.
LetA∈Qm×m, B∈Qn×n. Then(1)
f(I,Aσ)(λ)is a real polynomial, andf(I,Aσ)(λ)=∑k=02m‍a2kλ2k;(2)
hAσ(λ)is a real polynomial, andhAσ(λ)=∑k=02m‍a2kλ2(2m-k);(3)
hAσ(Bσ)=(gAσ(BB~))σPn,f(I,Aσ)(Bσ)=(pAσ(BB~))σPn,in whichgAσ(λ)=∑k=02m‍a2kλm-k, pAσ(λ)=∑k=02m‍a2kλkare real polynomials.Proof.
By Proposition4, we easily know that ak is a real number, and a2k+1=0. For any k, by Proposition 3, we have Bσ2k=((BB~)k)σPn, so we can obtain the result (3).


## 2.3. On Solutions to the Quaternionj-Conjugate Matrix Equation X-AX^B=CY

In this subsection, we discuss the solution of the following quaternion matrix equation:(17)X-AX^B=CY,
by means of real representation, where A∈Qn×n, B∈Qp×p, and C∈Qn×r are known matrices, X∈Qn×p and Y∈Qr×p are unknown matrices.We first define the real representation of quaternion matrix equation (17) by
(18)V-AσVBσ=CσPrW.According to (1) in Proposition3, the quaternion matrix equation (17) is equivalent to the following equation:
(19)(X-AX^B)σ=Xσ-AσXσBσ.
Therefore, the matrix equation (17) can be converted into
(20)Xσ-AσXσBσ=CσPrYσ.
Thus, we have the following conclusion.Proposition 6.
Given the quaternion matricesA∈Qn×n, B∈Qp×p and C∈Qn×r, then the quaternion matrix equation (17) has a solution (X,Y) if and only if the real representation matrix equation (18) has a solution (V,W)=(Xσ,Yσ).Theorem 7.
LetA∈Qn×n, B∈Qp×p, and C∈Qn×r. Then quaternion matrix equation (17) has a solution (X,Y) if and only if real representation matrix equation (18) has a solution (V,W). Furthermore, if (V,W) is a solution to (18), then the following quaternion matrices are solutions to quaternion matrix equation (17):
(21)X=116[IniInjInkIn]×(V-Qn-1VQp+Rn-1VRp-Sn-1VSp)[Ip-iIp-jIp-kIp],Y=116[IriIrjIrkIr]×(W-Qn-1WQp+Rn-1WRp-Sn-1WSp)[Ip-iIp-jIp-kIp].Proof.
By (3) of Proposition3, the quaternion matrix equation (18) is equivalent to
(22)V-Rn-1AσRnVRp-1BσRp=Rn-1CσRrPrW.
After multiplying the two sides of quaternion matrix equation (22) by Rp-1, we can obtain
(23)VRp-1-Rn-1AσRnVRp-1Bσ=Rn-1CσRrPrWRp-1.
Before multiplying the two sides of quaternion matrix equation (23) by Rn, we have
(24)RnVRp-1-AσRnVRp-1Bσ=CσRrPrWRp-1.
Noting that  Rp-1=-Rp, RrPr=PrRr, we give
(25)Rn-1VRp-AσRn-1VRpBσ=CσPrRr-1WRp.
This shows that if (V,W) is a real solution of matrix equation (18), then (Rn-1VRp,Rr-1WRp) is also a real solution of quaternion matrix equation (18). In addition, according to (3) of Proposition 3, the quaternion matrix equation (18) is also equivalent to
(26)V-QnAσQnVQpBσQp=QnCσQrPrW.
After multiplying the two sides of quaternion matrix equation (26) by Qp-1, we have
(27)VQp-1-QnAσQnVQpBσ=QnCσQrPrWQp-1.
Noting that  Qp-1=-Qp, QrPr=-PrQr, before multiplying the two sides of the quaternion matrix equation (27) by Qn-1, gives
(28)(-Qn-1VQp)-Aσ(-Qn-1VQp)Bσ=CσPp(-Qr-1WQp).
This is to say that if (V,W) is a real solution of matrix equation (18), then (-Qn-1VQp,-Qr-1WQp) is also a real solution of matrix equation (18). Similarly, we can prove that (-Sn-1VSp,-Sr-1WSp) is also a real solution of quaternion matrix equation (18). In this case, the conclusion can be obtained along the line of the proof of Theorem  4.2 in [13].Theorem 8.
Given the quaternion matricesA∈Qn×n, B∈Qp×p, and C∈Qn×r, let
(29)f(I,Aσ)(s)=det(I4n-sAσ)=∑k=02na2ks2k,pAσ(s)=∑k=02na2ksk.
Then the matrices X∈Qn×p, Y∈Qr×p are given by
(30)X=∑k=02n-1∑s=k2n-1α2k(AA^)s-kCZ(B^B)s+∑k=02n-1∑s=k2n-1α2k(AA^)s-kAC^Z^B(B^B)s,Y=ZpAσ(B^B),
in which Z is an arbitrary quaternion matrix.Proof.
If Yakubovich quaternionj-conjugate matrix equation (17) has solution (X,Y), then real representation matrix equation (18) has solution (V,W)=(Xσ,Yσ) with the free parameter Zσ. By Theorems 2 and 7, we have
(31)Xσ=∑k=02n-1∑j=0kαjAσj-kCσPrZσBσj=∑k=02n-1∑j=2k4n-1α2kAσj-2kCσPrZσBσj=∑k=02n-1α2k[∑s=k2n-1Aσ2s-2kCσPrZσBσ2s+∑s=k2n-1‍Aσ2s-2k+1CσPrZσBσ2s+1]=∑k=02n-1α2k×[∑s=k2n-1((AA^)s-k)σPnCσPrZσ((B^B)s)σPp+∑s=k2n-1((AA^)s-k)σPnAσCσPrZσBσ((B^B)s)σPp]=∑k=02n-1α2k[∑s=k2n-1((AA^)s-kCZ(B^B)s)σ+∑s=k2n-1((AA^)s-kAC^Z^B(B^B)s)σ].
In addition, by Proposition 5, f(I,Aσ)(s) is a real polynomial and f(I,Aσ)(Bσ)=(pAσ(BB^))σPp. So according to Proposition 3, we obtain
(32)Yσ=Zσf(I,Aσ)(Bσ)=Zσ(pAσ(BB^))σPp=(ZpAσ(B^B))σ.
Thus, the conclusion above has been proved.In the following, we provide an equivalent statement of Theorem8.Theorem 9.
Given quaternion matricesA∈Qn×n, B∈Qp×p, and C∈Qn×r, let
(33)f(I,Aσ)(s)=det(I4n-sAσ)=∑k=02na2ks2k,pAσ(s)=∑k=02na2ksk.
Then the matrices X∈Qn×p, Y∈Qr×p given by (30) have the following equivalent form:
(34)X=Qc(AA^,C,2n)Sr(I,Aσ)Qo(BB^,Z,2n)+Qc(AA^,AC^,2n)Sr(I,Aσ)Qo(B^B,Z^B,2n),Y=ZpAσ(B^B),
in which Z is an arbitrary quaternion matrix.Proof.
By the direct computation, we have(35)∑k=02n-1∑s=k2n-1α2k(AA^)s-kCZ(B^B)s=Qc(AA^,C,n)Sr(I,Aσ)Qo(BB^,Z,2n),∑k=02n-1∑s=k2n-1α2k(AA^)s-kAC^Z^B(B^B)s=Qc(AA^,AC^,2n)Sr(I,Aσ)Qo(B^B,Z^B,2n).
Thus, the first conclusion has been proved. With this the second conclusion is obviously true.Finally, we consider the solution to the so-called Kalman-Yakubovichj-conjugate quaternion matrix equation
(36)X-AX^B=C.
Based on the main result proposed above, we have the following conclusions regarding the matrix equation (36).Corollary 10.
Given quaternion matricesA∈Qn×n, B∈Qp×p, and C∈Qn×p, let
(37)f(I,Aσ)(s)=det(I4n-sAσ)=∑k=02na2ks2k,pAσ(s)=∑k=02na2ksk.
If X is a solution of equation (36), then
(38)XpAσ(B^B)=∑k=02n-1∑j=k2n-1α2k(AA^)j-kC(B^B)j+∑k=02n-1∑j=k2n-1(AA^)j-kAC^B(B^B)j.Proof.
IfX is a solution of equation (36), then Y=Xσ is a solution of the equation Xσ-AσXσBσ=Cσ. By Theorem  3 in [22] and Proposition 3, we have
(39)Xσf(I,Aσ)(Bσ)=∑k=02n-1∑j=2k4n-1α2kAσj-2kCσBσj.
By Proposition 5, f(I,Aσ)(s) is a real polynomial and f(I,Aσ)(Bσ)=(pAσ(BB^))σPp. So from Proposition 3 and (39), we have
(40)[XpAσ(BB^)]σ=Xσ[pAσ(B^B)]σPp=Xσf(I,Aσ)(Bσ)=∑k=02n-1∑j=2k4n-1α2kAσj-2kCσBσj=∑k=02n-1α2k[∑j=k2n-1Aσ2j-2kCσBσ2j+∑j=k2n-1Aσ2j+1-2kCσBσ2j+1]=∑k=02n-1α2k[∑j=k2n-1((AA^)j-k)σPnCσ((B^B)j)σPp+∑j=k2n-1((AA^)j-k)σPnAσCσBσ((B^B)j)σPp]=∑k=02n-1α2k×[∑j=k2n-1((AA^)j-kC(B^B)j)σ+∑j=k2n-1((AA^)j-kAC^B(B^B)j)σ]=∑k=02n-1∑j=k2n-1α2k((AA^)j-kC(B^B)j)σ+∑k=02n-1∑j=k2n-1((AA^)j-kAC^B(B^B)j)σ.
Thus, the first conclusion has been proved. With this the second conclusion is obviously true.In the following, we provide an equivalent statement of Theorem7.Corollary 11.
Given quaternion matricesA∈Qn×n, B∈Qp×p, and C∈Qn×p, let
(41)f(I,Aσ)(s)=det(I4n-sAσ)=∑k=02na2ks2k,pAσ(s)=∑k=02na2ksk.
If X is a solution of (36), then
(42)XpAσ(B^B)=Qc(AA^,C,2n)Sp(I,Aσ)Qo(B^B,Ip,2n)+Qc(AA^,A,2n)Sn(I,Aσ)Qo(B^B,C^B,2n).


## 3. Complex Conjugate Matrix EquationX-AX¯B=CY

In this section, we study the solution to the complex matrix equation(43)X-AX¯B=CY,
where A∈Cn×n, B∈Cp×p, and C∈Cn×r. Next, we define real representation of complex matrix as follows.For any complex matrixA=A1+A2i∈Cm×n, Al∈Rm×n(l=1,2.), we define a real representation of a complex matrix as
(44)Aσ=[A1A2A2-A1]∈R2m×2n.
Then the real matrix Aσ is called real representation of complex matrix A.Let(45)Pt=[It00-It],Qt=[0It-It0],
in which It is t×t identity matrix. Then Pt, Qt are unitary matrices. The real presentation has the following properties, which are given by Jiang and Wei [14].Proposition 12.
Consider the following.(1)
IfA,B∈Cm×n,a∈R, then(A+B)σ=Aσ+Bσ,(aA)σ=aAσ,PmAσPn=(A¯)σ;(2)
letA∈Cm×n,C∈Cn×s,a∈R, then(AC)σ=AσPnCσ;(3)
ifA∈Cm×m, then complex matrixAis nonsingular if and only ifAσis nonsingular;(4)
ifA∈Cm×m, thenAσ2k=((AA¯)k)σPm;(5)
ifA∈Cm×n, thenQmAσQn=Aσ.Actually, since complex matrix is a special case of quaternion matrix, in this case, we also have the following similar results. Because the proofs are similar to Section2 and are omitted.Theorem 13.
Given complex matricesA∈Cn×n, B∈Cp×p, and C∈Cn×r. Let
(46)f(I,Aσ)(s)=det(I2n-sAσ)=∑k=0na2ks2k,pAσ(s)=∑k=0na2ksk.
Then the solution to the matrix equation (43) is given by
(47)X=∑k=0n-1∑s=kn-1α2k(AA¯)s-kCZ(B¯B)s+∑k=0n-1∑s=kn-1α2k(AA¯)s-kAC¯Z¯B(B¯B)s,Y=ZpAσ(B¯B).In the following, we provide an equivalent statement of Theorem13.Theorem 14.
Given complex matricesA∈Cn×n, B∈Cp×p, and C∈Cn×p, let
(48)f(I,Aσ)(s)=det(sI2n-Aσ)=∑k=0na2ks2k,pAσ(s)=∑k=0na2ksk.
Then the matrices X and Y given by (47) have the following equivalent form:
(49)X=Qc(AA¯,C,n)Sr(I,Aσ)Qo(B¯B,Z,n)+Qc(AA¯,AC¯,n)Sr(I,Aσ)Qo(B¯B,Z¯B,n),Y=ZpAσ(B¯B).Finally, we consider the solution to the so-called Kalman-Yakubovich-conjugate matrix(50)X-AX¯B=C.Based on the main result proposed above, we have the following conclusions regarding matrix equation (50).Theorem 15.
Given the complex matricesA∈Cn×n, B∈Cp×p, and C∈Cn×p, let
(51)f(I,Aσ)(s)=det(sI2n-Aσ)=∑k=0na2ks2k,pAσ(s)=∑k=0na2ksk.(1) If X is a solution of (50), then
(52)XpAσ(B¯B)=∑k=0n-1∑j=kn-1α2k(AA¯)j-kC(B¯B)j+∑k=0n-1∑j=kn-1α2k(AA¯)j-kAC¯B(B¯B)j.(2) If X is the unique solution of (50), then
(53)X=[∑k=0n-1∑j=kn-1α2k(AA¯)j-kC(B¯B)j+∑k=0n-1∑j=kn-1α2k(AA¯)j-kAC¯B(B¯B)j]×[pAσ(B¯B)]-1.Theorem 16.
Given the complex matricesA∈Cn×n, B∈Cp×p, and C∈Cn×p, let
(54)f(I,Aσ)(s)=det(sI2n-Aσ)=∑k=0na2ks2k,pAσ(s)=∑k=0na2ksk.(1) If X is a solution of (50), then
(55)XpAσ(B¯B)=Qc(AA¯,C,n)Sp(I,Aσ)Qo(B¯B,Ip,n)+Qc(AA¯,A,n)Sn(I,Aσ)Qo(B¯B,C¯B,n).(2) If X is the unique solution of (50), then
(56)X=[Qc(AA¯,C,n)Sp(I,Aσ)Qo(B¯B,Ip,n)+Qc(AA¯,A,n)Sn(I,Aσ)Qo(B¯B,C¯B,n)]×[pAσ(B¯B)]-1.


## 4. Illustrative Example

In this section, we give an example to obtain the solution of complex conjugate matrix equationX-AX¯B=CY.Example 1.
Consider Yakubovich-conjugate matrix equation in the form of (43) with the following parameters:
(57)A=[1+i2i40],B=[34+i1-2i],C=[32i2-i4].
According to the definition of real representation of a complex matrix, we have(58)Aσ=[1012400012-1000-40].
By some simple computations, we have
(59)f(I,Aσ)(λ)=64λ4-2λ2+1,pAσ(λ)=64λ2-2λ+1,S2(Aσ)=[I22I20I2],I2=[1001],Qc(AA¯,C,2)=[32i8+18i-8-4i2-i44-28i8-24i],Qo(B¯B,Z,2)=[1i-1111+2i9+3i-10+3i-2+6i],Qc(AA¯,AC¯,2)=[1+7i2+6i-30-2i-60+12i12-8i32-72i-32+16i],Qo(B¯B,Z¯B,2)=[3-i2+i-2-4-3i42-9i40-15i-32-15i-49-18i].
Choose
(60)Z=[1i-11],
then it follows from Theorem 14 that the solution of (43) is
(61)X=[659+840i1649+1118i1350-3683i1611-4132i],Y=[10603+2684i12078-133i-9261+4026i-6843+8052i].


## 5. Conclusions

In the present paper, by means of the real representation of a quaternion matrix, we study the quaternion matrix equationX-AX~B=CY. Compared to our previous results [10], there are no requirements on the coefficient matrix A. Explicit solutions to this quaternion matrix equation are established by application of the real representation of a quaternion matrix. As a special case of quaternion j-conjugate matrix equation, complex conjugate matrix equation X-AX¯B=CY is also considered and the explicit solutions to complex conjugate are proposed. In addition, the equivalent forms of the explicit solutions are given.



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