An Iterative Algorithm for the Least Squares Generalized Reflexive Solutions of the Matrix Equations AXB E,CXD F

and Applied Analysis 3 to a class of complex matrix equations with conjugate and transpose of the unknowns. Jonsson and Kågström 24, 25 proposed recursive block algorithms for solving the coupled Sylvester matrix equations and the generalized Sylvester and Lyapunov Matrix equations. Very recently, Huang et al. 26 presented a finite iterative algorithms for the one-sided and generalized coupled Sylvester matrix equations over generalized reflexive solutions. Yin et al. 27 presented a finite iterative algorithms for the two-sided and generalized coupled Sylvester matrix equations over reflexive solutions. For more studies on the matrix equations, we refer to 1–4, 16, 17, 28–40 . However, the problem of finding the least squares generalized reflexive solution of the matrix equation pair has not been solved. The following notations are also used in this paper. Let Rm×n denote the set of all m×n real matrices. We denote by the superscript T the transpose of a matrix. In matrix space Rm×n, define inner product as 〈A,B〉 trace BA for all A,B ∈ Rm×n, and ‖A‖ represents the Frobenius norm of A. R A represents the column space of A. vec · represents the vector operator, that is, vec A aT1 , a T 2 , . . . , a T n T ∈ Rmn for the matrix A a1, a2, . . . , an ∈ Rm×n, ai ∈ Rm, i 1, 2, . . . , n. A ⊗ B stands for the Kronecker product of matrices A and B. This paper is organized as follows. In Section 2, we will solve Problem 1 by constructing an iterative algorithm, that is, for an arbitrary initial matrix X1 ∈ Rm×n r P,Q , we can obtain a solution X∗ ∈ Rm×n r P,Q of Problem 1 within finite iterative steps in the absence of round-off errors. The convergence of the algorithm is also proved. Let X1 AHB CĤD PAHBQ PCĤDQ, where H ∈ Rp×q, Ĥ ∈ Rs×t are arbitrary matrices, or more especially, let X1 0 ∈ Rm×n r P,Q ; we can obtain the unique least-norm solution X∗ of Problem 1. Then in Section 3, we give the optimal approximate solution of Problem 2 by finding the least-norm generalized reflexive solution of a corresponding new minimum Frobenius norm residual problem. In Section 4, several numerical examples are given to illustrate the application of our iterative algorithm. 2. Solution of Problem 1 In this section, we firstly introduce some definitions, lemmas, and theorems which are required for solving Problem 1. Then we present an iterative algorithm to obtain the solution of Problem 1. We also prove that it is convergent. The following definitions and lemmas come from 41 , which are needed for our derivation. Definition 2.1. A set of matrices S ∈ Rm×n is said to be convex if for X1, X2 ∈ S and α ∈ 0, 1 , αX1 1 − α X2 ∈ S. Let Rc denote a convex subset of Rm×n. Definition 2.2. A matrix function f : Rc → R is said to be convex if f αX1 1 − α X2 ≤ αf X1 1 − α f X2 2.1 for X1, X2 ∈ Rc and α ∈ 0, 1 . Definition 2.3. Let f : Rc → R be a continuous and differentiable function. The gradient of f is defined as ∇f X ∂f X /∂xij . 4 Abstract and Applied Analysis Lemma 2.4. Let f : Rc → R be a continuous and differentiable function. Then f is convex on Rc if and only if f Y ≥ f X ∇f X , Y −X 2.2