By the complex conjugate matrix equation it means a complex matrix equation with the conjugate of the unknown matrices. The earliest investigated complex conjugate matrix equation may be the following matrix equation with unknown matrixX:
AX−XB=C, (1.39)
which was studied by Bevis, Hall and Hartwing in [21]. This equation can be viewed as a conjugate version of the normal Sylvester matrix equation (1.8). Due to this reason, in this book the matrix equation (1.39) will be referred to as normal con- Sylvester matrix equation. In [21, 142], a necessary and sufficient condition for the existence of a solution to the normal con-Sylvester equation was established by using the concept of consimilarity [141, 148] associated with two partitioned matrices related to A, B, and C. The general solution to the corresponding homogeneous equation AX−XB = 0 was given in [21], where it was shown that any solution to the normal con-Sylvester equation can be written as a particular solution plus a complementary solution to the homogeneous equation. In [20], a solution to the normal con-Sylvester matrix equation (1.39) was obtained in the case where the matricesAandBare both in consimilarity Jordan forms. In [258] explicit solutions expressed by the original coefficient matrices were established with the help of a kind of real representation [155] of complex matrices based on a proposed solution to the normal Sylvester equation. Different from [20], the solution can be obtained in terms of original coefficient matrices instead of real representation matrices by using a developed result on characteristic polynomial of real representation matrices. It was
also shown in [258] that the solution to the normal con-Sylvester matrix equation (1.39) is unique if and only ifAAandBBhave no common eigenvalues. In addition, the homogeneous equationAX−XB=0 was also investigated in [225], and it was shown that the solution can be obtained in terms of the Kronecker canonical form of the matrix pencilB+sAand the two non-singular matrices that transform this pencil into its Kronecker canonical form.
The following conjugate version of the Kalman-Yakubovich matrix equation (1.9) was firstly investigated in [155]:
X−AXB=C. (1.40)
Similarly, the matrix equation (1.40) will be referred to as con-Kalman-Yakubovich matrix equation in this book. In [155] with the aid of a real representation of complex matrices, the solvability conditions and solutions of this equation were established in terms of its real representation matrix equation. Based on a result on character- istic polynomials of the real representation matrix for a complex matrix, an explicit solution of this equation was given for the con-Kalman-Yakubovich matrix equation (1.40) in terms of controllability and observability matrices in [280], while in [308]
the con-Kalman-Yakubovich equation was transformed into the Kalman-Yakubovich equation. In [264], some Smith-type iterative algorithms were developed, and the corresponding convergence analysis was also given. A general form of the matrix equations (1.39) and (1.40) is the following equation
EXF−AX =C,
where X is the matrix to be determined, andE,A,F, and C are known complex matrices with E,A, andF being square. This equation was investigated in [242], and three approaches were provided to obtain the solution of this equation. The first approach is to transform it into a real matrix equation with the help of real representations of complex matrices. In the second approach, the solution is given in terms of the characteristic polynomial of a constructed matrix pair. In the third approach, the solution can be neatly expressed in terms of controllability matrices and observability matrices.
Some conjugate versions of the matrix equations (1.20), (1.22), and (1.25) have also been investigated in literature. In [269], the following con-Yakubovich matrix equation was considered:
X−AXF=BY,
whereXandYare the unknown matrices. With the aid of a relation on the characteris- tic polynomial of the real representation for a complex matrix, some explicit paramet- ric expressions of the solutions to this equation were derived. The con-Yakubovich matrix equation was also investigated in [266]. Different from the approach in [269], the solutions of this equation were derived in [266] beyond the framework of real representations of complex matrices. With Smith normal form reduction as a tool, a general complete parametric solution was proposed in terms of the original matri- ces. This solution is expressed in a finite series form, and can offer all the degrees
of freedom which are represented by a free parameter matrixZ. In [268, 273], the following con-Sylvester matrix equations were investigated:
AX+BY =XF, AX+BY =XF+R,
and the solutions to these two equations were explicitly given. In [268], the solution was obtained based on the Leverrier algorithm. While in [273], the general solution was given by solving a standard linear equation. In [282], the following generalized con-Sylvester matrix equation was considered:
AX+BY =EXF,
and two approaches were established to solve this matrix equation. The first approach is based on the real representation technique. The basic idea is to transform it into the generalized Sylvester matrix equation. In the second approach the solution to this matrix equation can be explicitly provided. One of the obtained solutions in [282] is expressed in terms of the controllability matrix and the observability matrix. Such a feature may bring much convenience in some analysis and design problems related to the generalized con-Sylvester matrix equation. In [252], an iterative algorithm was presented for solving the nonhomogeneous generalized con-Sylvester matrix equation
AX+BY =EXF+R.
This iterative method in [252] can give an exact solution within finite iteration steps for any initial values in the absence of round-off errors. Another feature of the proposed algorithm is that it is implemented by original coefficient matrices. In [267], the so-called con-Sylvester-polynomial matrix equation, which is a general form of the con-Sylvester and con-Yakubovich matrix equations, was investigated.
By using the conjugate product of complex polynomial matrices proposed in [271] as a tool, the complete parametric solution was given for the con-Sylvester-polynomial matrix equation.
In [241], the following matrix equation was investigated
AX−XHB=C, (1.41)
whereA∈Cm×n,B∈Cn×m, andC ∈Cm×mare known matrices, andX ∈Cn×mis the matrix to be determined. It was pointed out in [241] that this matrix equation has a solution if and only if there exists a nonsingular matrixS∈C(n+m)×(n+m)such that
0 −A B 0
=S C−A
B 0
SH.
In [48], an alternative proof was provided for the result on the matrix equation (1.41) in [241]. For the matrix equation (1.41) with m = n, a necessary and sufficient
condition was given in [168] for the existence of its unique solution. In addition, an efficient numerical algorithm was also given in [48] to solve the matrix equation (1.41) by using the generalized Schur decomposition.
For some more general complex conjugate matrix equations, there are results available in literature. In [281], an iterative algorithm was constructed to solve the so-called extended con-Sylvester matrix equations by using the hierarchical identifi- cation principle. Such an idea was generalized to solve some other equations in [219, 265, 272].
Another numerical approach was the finite iterative algorithm. In [275], a finite iterative algorithm was proposed to solve the so-called extended con-Sylvester matrix equations by using a newly defined real inner product as a tool. The real inner product in [275] was extended in [270] to the case of matrix groups. With the help of this inner product, a finite iterative algorithm was established in [270] to solve a class of coupled con-Sylvester matrix equations. The real inner products in [270, 275] were also used to solve some other complex matrix equations in [204, 274].