An Iterative Algorithm for the Generalized Reflexive Solution of the Matrix Equations A X B = E, C X D = F

An iterative algorithm is constructed to solve the linear matrix equation pair 𝐴𝑋𝐵=𝐸,𝐶𝑋𝐷=𝐹 over generalized reflexive matrix 𝑋. When the matrix equation pair 𝐴𝑋𝐵=𝐸,𝐶𝑋𝐷=𝐹 is consistent over generalized reflexive matrix 𝑋, for any generalized reflexive initial iterative matrix 𝑋1, 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 𝐴𝑋𝐵=𝐸,𝐶𝑋𝐷=𝐹 for a given generalized reflexive matrix 𝑋0 can be derived by finding the least-norm generalized reflexive solution of a new corresponding matrix equation pair 𝐴𝐹𝑋𝐵=𝐸,𝐶𝑋𝐷= with 𝐸=𝐸−𝐴𝑋0𝐵,𝐹=𝐹−𝐶𝑋0𝐷. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

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