The general coupled linear matrix equations with conjugate and transpose unknowns over the mixed groups of generalized reflexive and anti-reflexive matrices

In this paper, we consider a class of general coupled linear matrix equations over the complex number field. The mentioned coupled linear matrix equations contain the unknown complex matrix groups X = (X-1, X-2,..., X-q) and Z = (Z(1), Z(2),..., Z(q)). The conjugate and transpose of the unknown matrices X-i and Z(i), i is an element of I [1, q], appear in the considered coupled linear matrix equations. An iterative algorithm is presented to determine the unknown matrix groups X and Z such that X and Z are the groups of the generalized reflexive and anti-reflexive matrices, respectively. The proposed algorithm determines the solvability of the general coupled linear matrix equations over the generalized reflexive and anti-reflexive matrices, automatically. When the general coupled linear matrix equations are consistent over the generalized reflexive and anti-reflexive matrices, it is shown that the algorithm converges within finite number of steps, in the exact arithmetic. In addition, the optimal approximately generalized reflexive and anti-reflexive solution groups to the given arbitrary matrix groups Gamma(x) = (Gamma(1x), Gamma(2x),..., Gamma(qx)) and Gamma(z) = (Gamma(1z), Gamma(2z),..., Gamma(qz)) are derived. Finally, some numerical results are given to illustrate the validity of the presented theoretical results and feasibly of the proposed algorithm.