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

and Applied Analysis 3 In [21], we can find the following well-known generalized Faddeev-Leverrier algorithm: R k = R k−1 A + α k I n , R 0 = I n , k = 1, 2, . . . , n, α k = trace (R k−1 A) k , α 0 = 1, k = 1, 2, . . . , n, (8) where α i , i = 0, 1, 2, . . . , n − 1, are the coefficients of the characteristic polynomial of the matrix A, and R i , i = 0, 1, . . . , n−1, are the coefficientmatrices of the adjointmatrix adj(sI n − A). Theorem 2. Given matrices A ∈ R, B ∈ R, C ∈ R, let f (I,A) (s) = det (I − sA) = αns n + ⋅ ⋅ ⋅ + α 1 s + α 0 , α 0 = 1. (9) Then the matrices X and Y given by (4) have the following equivalent form:


Introduction
The linear matrix equation  −  = , 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 -conjugate matrix equation − X =  (where X denotes the -conjugate of quaternion matrix ) 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   +   =  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 conjugate matrix equation  −  X = ,  −  X = , but here the known quaternion matrix  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  or the Leverrier algorithm.The existence of solution to the matrix equation  −  = , 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 ∑  =0     =  over quaternion field are obtained [15].The authors in [16][17][18] have provided the consistence of the matrix equation  −  =  via the consimilarity of two matrices.In [19], Wu et al. construct some explicit expressions of the solution of the matrix equation  −  =  by means of a real representation of a complex matrix.It is shown that there exists a unique solution if and only if  and  have no common eigenvalues.
In this paper, we study quaternion -conjugate matrix equation  −  X =  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 .We propose the explicit solutions to the above Yakubovich--conjugate quaternion matrix equation.As the special case of quaternion -conjugate matrix equation  −  X = , complex conjugate matrix equation  −  =  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.Let  denote the real number field,  the complex number field, and  =  ⊕  ⊕  ⊕  the quaternion field, where denotes the set of all  ×  matrices on  ( or ).For any matrix  ∈  × ,   , ,   , det , and  * represent the transpose, conjugate, conjugate transpose, determinant, and adjoint of , respectively.In addition, symbol   is the real representation of quaternion matrix .⊗ = (  ) denotes the Kronecker product of two matrices  and .
Then, all the solutions to the Yakubovich matrix equation (2) can be established as where the matrix  ∈  × is an arbitrary matrix.
Proof.We first show that the matrices  and  given in (4) are solutions of the matrix equation (2).By the direct calculation we have Due to the relation ( − )adj( − ) =  det( − ), it is easily derived that So one has Thus, the matrices  and  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  degrees of freedom in the solution of matrix equation (2), while solution (4) has exactly  parameters represented by the elements of the free matrix .Therefore, in the following we only need to show that all the parameters in the matrix  contribute to the solution.To do this, it suffices to show that the mapping  → (,) defined by ( 5) is injective.This is true since  (,) () is nonsingular under the condition of { | det( − ) = 0} ∩ () = .The proof is thus completed.
This relation can be compactly expressed as Substituting this into the expression of  in (10) and recording the sum, we have Combining this with Theorem 1 gives the conclusion.

Real Representation of a Quaternion Matrix. For any quaternion matrix
∈  × ( = 1, 2, 3, 4), the real representation matrix of quaternion matrix  can be defined as For a  ×  quaternion matrix , we define    = (  )  .In addition, if we let in which   is a  ×  identity matrix, then   ,   ,   ,   are unitary matrices.
The real representation has the following properties, which are given in [13].
For any  ∈  × , let the characteristic polynomial of the real representation matrix   be (2) ℎ   () is a real polynomial, and Proof.By Proposition 4, we easily know that   is a real number, and  2+1 = 0.For any , by Proposition 3, we have  2  = (( B)  )    , so we can obtain the result (3).

On Solutions to the Quaternion 𝑗-Conjugate Matrix
Equation − X = .In this subsection, we discuss the solution of the following quaternion matrix equation: by means of real representation, where  ∈  × ,  ∈  × , and  ∈  × are known matrices,  ∈  × and  ∈  × are unknown matrices.We first define the real representation of quaternion matrix equation (17) by According to (1) in Proposition 3, the quaternion matrix equation ( 17) is equivalent to the following equation: Therefore, the matrix equation ( 17) can be converted into Thus, we have the following conclusion.
Proof.By (3) of Proposition 3, the quaternion matrix equation ( 18) is equivalent to After multiplying the two sides of quaternion matrix equation ( 22) by  −1  , we can obtain Before multiplying the two sides of quaternion matrix equation (23) by   , we have Noting that  −1  = −  ,     =     , we give This shows that if (, ) is a real solution of matrix equation (18), then ( −1    ,  −1    ) 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 After multiplying the two sides of quaternion matrix equation (26) by  −1  , we have Noting that  −1  = −  ,     = −    , before multiplying the two sides of the quaternion matrix equation ( 27) by  −1  , gives This is to say that if (, ) is a real solution of matrix equation (18), then (− −1    , − −1    ) is also a real solution of matrix equation (18).Similarly, we can prove that (− −1    , − −1    ) 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].
In the following, we provide an equivalent statement of Theorem 8.

Theorem 9.
Given quaternion matrices  ∈  × ,  ∈  × , and  ∈  × , let Then the matrices  ∈  × ,  ∈  × given by (30) have the following equivalent form: Proof.By the direct computation, we have 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-Yakubovich -conjugate quaternion matrix equation Based on the main result proposed above, we have the following conclusions regarding the matrix equation (36).
Corollary 10.Given quaternion matrices  ∈  × ,  ∈  × , and  ∈  × , let If  is a solution of equation (36), then Proof.If  is a solution of equation ( 36), then  =   is a solution of the equation   −       =   .By Theorem 3 in [22] and Proposition 3, we have By Proposition 5,  (,  ) () is a real polynomial and  (,  ) (  ) = (   ( B))    .So from Proposition 3 and (39), we have Thus, the first conclusion has been proved.With this the second conclusion is obviously true.
In the following, we provide an equivalent statement of Theorem 7.

Complex Conjugate Matrix Equation 𝑋−𝐴𝑋𝐵 = 𝐶𝑌
In this section, we study the solution to the complex matrix equation where  ∈  × ,  ∈  × , and  ∈  × .Next, we define real representation of complex matrix as follows.
For any complex matrix  =  1 +  2  ∈  × ,   ∈  × ( = 1, 2.), we define a real representation of a complex matrix as Then the real matrix   is called real representation of complex matrix .Let in which   is  ×  identity matrix.Then   ,   are unitary matrices.The real presentation has the following properties, which are given by Jiang and Wei [14].
Proposition 12. Consider the following.
( 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 Section 2 and are omitted.

Illustrative Example
In this section, we give an example to obtain the solution of complex conjugate matrix equation  −  = .] . (57) According to the definition of real representation of a complex matrix, we have (61)

Conclusions
In the present paper, by means of the real representation of a quaternion matrix, we study the quaternion matrix equation  −  X = .Compared to our previous results [10], there are no requirements on the coefficient matrix .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 -conjugate matrix equation, complex conjugate matrix equation  −  =  is also considered and the explicit solutions to complex conjugate are proposed.In addition, the equivalent forms of the explicit solutions are given.