## An efficient algorithm for the least-squares reflexive solution of the matrix equation $$A_{1}XB_{1} = C_{1}, A_{2}XB_{2} = C_{2}$$.(English)Zbl 1115.65048

In this paper, an iterative method for solving the minimum Frobenius norm residual problem $\left\|\begin{pmatrix} A_1 XB_1 \\ A_2 XB_2 \end{pmatrix}- \begin{pmatrix} C_1 \\ C_2 \end{pmatrix}\right\|=\min$ with an unknown reflexive matrix $$X$$ with respect to a generalized reflection matrix $$P$$ is introduced, where the matrices $$P$$ and $$X$$ satisfy $$P^T=P$$, $$P^2=I$$ and $$X=XPX$$ by definition. With any initial reflexive matrix $$X_1$$, the matrix sequence $$\{X_k \}$$ converges to its solution within at most $$n^2$$ steps, theoretically. In addition, if $X_1=A_1^T H_1 B_1^T + PA_1^T H_1 B_1^T P + A_2^T H_2 B_2^T + PA_2^T H_2 B_2^T P$ is used for the initial reflexive matrix with arbitrary matrices $$H_1,H_2$$, the solution is the least Frobenius norm solution. The numerical experiments support theoretical results.

### MSC:

 65F30 Other matrix algorithms (MSC2010) 15A24 Matrix equations and identities
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### References:

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