Article

# Linear restriction problem of Hermitian reflexive matrices and its approximation

College of Mathematics and Econometrics, Hunan University, Changsha 410082, PR China

Applied Mathematics and Computation (Impact Factor: 1.55). 06/2008; 200(1):341-351. DOI: 10.1016/j.amc.2007.11.020 Source: DBLP

**ABSTRACT**

In this paper, we consider a linear restriction problem of Hermitian reflexive matrices and its approximation. By using the properties and structure of Hermitian reflexive matrices and the special properties of reflexive vectors and anti-reflexive vectors, we convert the linear restriction problem to an equivalence problem trickily, which is a special feature of this paper and is a different method from other articles. Then we solve this problem completely and also obtain its optimal approximate solution. Moreover, an algorithm provided for it and the numerical examples show that the algorithm is feasible.

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**ABSTRACT:**In this paper, The Hermitian reflexive solutions and the anti-Hermitian reflexive solutions of matrix equations AX = B, XC = D are considered. With special properties of partitioned matrices and Hermitian reflexive (anti-Hermitian reflexive) matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is considered. - [Show abstract] [Hide abstract]

**ABSTRACT:**Let P and Q be two generalized reflection matrices, i.e, P=PH, P2=I and Q=QH, Q2=I. An n×n matrix A is said to be generalized reflexive (generalized anti-reflexive) with respect to the matrix pair (P;Q) if A=PAQ (A=-PAQ). It is obvious that any n×m matrix is also a generalized reflexive with respect to the matrix pair (In;Im). By extending the conjugate gradient least square (CGLS) approach, the present paper treats two iterative algorithms to solve the system of matrix equationsF1(X)=A1,F2(X)=A2, ⋮⋮⋮Fm(X)=Am,(including the Sylvester and Lyapunov matrix equations as special cases) over the generalized reflexive and anti-reflexive matrices, where F1,F2,⋯,Fm are the linear operators from Cn×n onto Cri×si and Ai∈Cri×si for i=1,2,⋯,m. When this system is consistent over the generalized reflexive (generalized anti-reflexive) matrix, it is proved that the first (second) iterative algorithm converges to a generalized reflexive (generalized anti-reflexive) solution for any initial generalized reflexive (generalized anti-reflexive) matrix. Also the first (second) iterative algorithm can obtain the the least Frobenius norm generalized reflexive (generalized anti-reflexive) solution for special initial generalized reflexive (generalized anti-reflexive) matrix. Furthermore, the optimal approximation generalized reflexive (generalized anti-reflexive) solution to a given generalized reflexive (generalized anti-reflexive) matrix can be derived by finding the least Frobenius norm generalized reflexive (generalized anti-reflexive) solution of a new system of matrix equations. Finally, we test the proposed iterative algorithms and show their effectiveness using numerical examples. - [Show abstract] [Hide abstract]

**ABSTRACT:**The main purpose of this correspondence is to establish two gradient based iterative (GI) methods extending the Jacobi and Gauss-Seidel iterations for solving the generalized Sylvester-conjugate matrix equation A1XB 1 + A2XB2 + C1YD1 + C2YD2 = E, over reflexive and Hermitian reflexive matrices. It is shown that the iterative methods, respectively, converge to the reflexive and Hermitian reflexive solutions for any initial reflexive and Hermitian reflexive matrices. We report numerical tests to show the effectiveness of the proposed approaches.

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