## An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations $$A_{1}XB_{1} = C_{1}, A_{2}XB_{2} = C_{2}$$.(English)Zbl 1134.65032

The symmetric solutions of the systems of matrix equations $$A_1XB_1=C_1$$, $$A_2XB_2=C_2$$ 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 $$A_1\hat XB_1=\hat C_1$$, $$A_2\hat XB_2=\hat C_2$$.
Finally, the author demonstrates the applicability of the proposed method on systems of matrix equations.

### MSC:

 65F30 Other matrix algorithms (MSC2010) 15A24 Matrix equations and identities 65F10 Iterative numerical methods for linear systems
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### References:

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