An Iterative Algorithm for the Generalized Reflexive Solution of the Matrix Equations ð ´ ð ‘‹ ð µ = ð ¸ , ð ¶ ð ‘‹ ð · = ð ¹

An iterative algorithm is constructed to solve the linear matrix equation pair ð ´ ð ‘‹ ð µ = ð ¸ , ð ¶ ð ‘‹ ð · = ð ¹ over generalized reflexive matrix ð ‘‹ . When the matrix equation pair ð ´ ð ‘‹ ð µ = ð ¸ , ð ¶ ð ‘‹ ð · = ð ¹ is consistent over generalized reflexive matrix ð ‘‹ , for any generalized reflexive initial iterative matrix ð ‘‹ 1 , the generalized reflexive solution can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors. The unique least-norm generalized reflexive iterative solution of the matrix equation pair can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate solution of ð ´ ð ‘‹ ð µ = ð ¸ , ð ¶ ð ‘‹ ð · = ð ¹ for a given generalized reflexive matrix ð ‘‹ 0 can be derived by finding the least-norm generalized reflexive solution of a new corresponding matrix equation pair ð ´ î ‚ î ‚ î ‚ î ‚ ð ¹ ð ‘‹ ð µ = ð ¸ , ð ¶ ð ‘‹ ð · = with î ‚ ð ¸ = ð ¸ âˆ’ ð ´ ð ‘‹ 0 î ‚ ð µ , ð ¹ = ð ¹ âˆ’ ð ¶ ð ‘‹ 0 ð · . Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.