On equivalence and explicit solutions of a class of matrix equations

It is shown in this paper that three types of matrix equations AX−XF=BY,AX−EXF=BY and which have wide applications in control systems theory, are equivalent to the matrix equation with their coefficient matrices satisfying some relations. Based on right coprime factorization to , explicit solutions to the equation are proposed and thus explicit solutions to the former three types of matrix equations can be immediately established. With the special structure of the proposed solutions, necessary conditions to the nonsingularity of matrix X are also obtained. The proposed solutions give an ultimate and unified formula for the explicit solutions to these four types of linear matrix equations.     

On conjugate product of complex polynomials

Some concepts, such as divisibility, coprimeness, in the framework of ordinary polynomial product are extended to the framework of conjugate product. Euclidean algorithm for obtaining greatest common divisors in the framework of conjugate product is also established. Some criteria for coprimeness are established.     

Weighted least squares solutions to general coupled Sylvester matrix equations

This paper is concerned with weighted least squares solutions to general coupled Sylvester matrix equations. Gradient based iterative algorithms are proposed to solve this problem. This type of iterative algorithm includes a wide class of iterative algorithms, and two special cases of them are studied in detail in this paper. Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented. Sufficient conditions that are easy to compute are also given. The optimal step sizes such that the convergence rates of the algorithms, which are properly defined in this paper, are maximized and established. Several special cases of the weighted least squares problem, such as a least squares solution to the coupled Sylvester matrix equations problem, solutions to the general coupled Sylvester matrix equations problem, and a weighted least squares solution to the linear matrix equation problem are simultaneously solved. Several numerical examples are given to illustrate the effectiveness of the proposed algorithms.     

The complete solution to the Sylvester-polynomial-conjugate matrix equations

In this paper we propose two new operators for complex polynomial matrices. One is the conjugate product and the other is the Sylvester-conjugate sum. Then some important properties for these operators are proved. Based on these derived results, we propose a unified approach to solving a general class of Sylvester-polynomial-conjugate matrix equations, which include the Yakubovich-conjugate matrix equation as a special case. The complete solution of the Sylvester-polynomial-conjugate matrix equation is obtained in terms of the Sylvester-conjugate sum, and such a proposed solution can provide all the degrees of freedom with an arbitrarily chosen parameter matrix.     

Least squares solution with the minimum-norm to general matrix equations via iteration

Two iterative algorithms are presented in this paper to solve the minimal norm least squares solution to a general linear matrix equations including the well-known Sylvester matrix equation and Lyapunov matrix equation as special cases. The first algorithm is based on the gradient based searching principle and the other one can be viewed as its dual form. Necessary and sufficient conditions for the step sizes in these two algorithms are proposed to guarantee the convergence of the algorithms for arbitrary initial conditions. Sufficient condition that is easy to compute is also given. Moreover, two methods are proposed to choose the optimal step sizes such that the convergence speeds of the algorithms are maximized. Between these two methods, the first one is to minimize the spectral radius of the iteration matrix and explicit expression for the optimal step size is obtained. The second method is to minimize the square sum of the F-norm of the error matrices produced by the algorithm and it is shown that the optimal step size exits uniquely and lies in an interval. Several numerical examples are given to illustrate the efficiency of the proposed approach.     

On solutions of matrix equations and

With the help of the Kronecker map, a complete, general and explicit solution to the Yakubovich matrix equation V−AVF=BW, with F in an arbitrary form, is proposed. The solution is neatly expressed by the controllability matrix of the matrix pair (A,B), a symmetric operator matrix and an observability matrix. Some equivalent forms of this solution are also presented. Based on these results, explicit solutions to the so-called Kalman–Yakubovich equation and Stein equation are also established. In addition, based on the proposed solution of the Yakubovich matrix equation, a complete, general and explicit solution to the so-called Yakubovich-conjugate matrix is also established by means of real representation. Several equivalent forms are also provided. One of these solutions is neatly expressed by two controllability matrices, two observability matrices and a symmetric operator matrix.     

Finite iterative solutions to coupled Sylvester-conjugate matrix equations

This paper is concerned with solutions to the so-called coupled Sylveter-conjugate matrix equations, which include the generalized Sylvester matrix equation and coupled Lyapunov matrix equation as special cases. An iterative algorithm is constructed to solve this kind of matrix equations. By using the proposed algorithm, the existence of a solution to a coupled Sylvester-conjugate matrix equation can be determined automatically. When the considered matrix equation is consistent, it is proven by using a real inner product in complex matrix spaces as a tool that a solution can be obtained within finite iteration steps for any initial values in the absence of round-off errors. Another feature of the proposed algorithm is that it can be implemented by using original coefficient matrices, and does not require to transform the coefficient matrices into any canonical forms. The algorithm is also generalized to solve a more general case. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

On matrix equations and

With the help of the concept of Kronecker map, an explicit solution for the matrix equation X−AXF=C is established. This solution is neatly expressed by a symmetric operator matrix, a controllability matrix and an observability matrix. In addition, the matrix equation is also studied. An explicit solution for this matrix equation is also proposed by means of the real representation of a complex matrix. This solution is neatly expressed by a symmetric operator matrix, two controllability matrices and two observability matrices.