A modified iterative algorithm for the (Hermitian) reflexive solution of the generalized Sylvester matrix equation

Recently, Ramadan et al. have focused on the following matrix equation:

A 1 V + A 2 V ¯ + B 1 W + B 2 W ¯ = E 1 V F 1 + E 2 V ¯ F 2 + C

and propounded two gradient-based iterative algorithms for solving the above matrix equation over reflexive and Hermitian reflexive matrices, respectively. In this paper, we develop two new iterative algorithms based on a two-dimensional projection technique for solving the mentioned matrix equation over reflexive and Hermitian reflexive matrices. The performance of our proposed algorithms is collated with the gradient-based iterative algorithms. It is both theoretically and experimentally demonstrated that the approaches handled surpass the offered algorithms in the earlier referred work in solving the mentioned matrix equation over reflexive and Hermitian reflexive matrices. In addition, it is briefly discussed that a one-dimensional projection technique can accelerate the speed of convergence of the gradient-based iterative algorithm for solving general coupled Sylvester matrix equations over reflexive matrices without assuming the restriction of the existence of a unique solution.