*(English)*Zbl 1134.65032

The symmetric solutions of the systems of matrix equations ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$ can not be easily obtained by applying matrix decompositions. The authors are proposing an iterative method to solve systems of matrix equations, where when the system of matrix equations is consistent, and its solution can be obtained within finite iterative steps, and its least-norm solution can be obtained by choosing a special kind of initial iterative matrix. Additionally, its optimal approximation solution to a given matrix can be derived by finding the least-norm symmetric solution of a new system of matrix equations ${A}_{1}\widehat{X}{B}_{1}={\widehat{C}}_{1}$, ${A}_{2}\widehat{X}{B}_{2}={\widehat{C}}_{2}$.

Finally, the author demonstrates the applicability of the proposed method on systems of matrix equations.

##### MSC:

65F30 | Other matrix algorithms |

15A24 | Matrix equations and identities |

65F10 | Iterative methods for linear systems |