Based on the method of the modified conjugate gradient to the linear matrix equation over constrained matrices, and by modifying the construction of some matrices, an iterative algorithm is presented to find the generalized reflexive solution of the matrix equation which is a special type with several matrix variables. The convergence of the iterative algorithm is proved. And the problem of the optimal approximation to the given matrix is solved in the generalized reflexive solution set of this matrix equation. When this matrix equation is consistent, its generalized reflexive solution can be obtained within finite iterative steps. And its least-norm generalized reflexive solution can be got by choosing the special initial matrices. The numerical example shows that the iterative algorithm is quite efficient.
Dehghan Mehdi, Hajarian Masoud. Two algorithms for finding the Hermitian reflexive and skew- Hermitian solutions of Sylvester matrix equations[J]. Appl. Math. Lett, 2011, 24: 444-449.
Wang Xiang, Wu Wuhua. A finite Iterative algorithm for Solving the generalized (P, Q)-reflexive solution of the linear systems of matrix equations[J]. Mathematical and Computer Modelling, 2011, 54: 2117-2131.