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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 $$\left\|\pmatrix A_1 XB_1 \\ A_2 XB_2 \endpmatrix- \pmatrix C_1 \\ C_2 \endpmatrix\right\|=\min$$ 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 15A24 Matrix equations and identities
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##### References:
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