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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 \[ \left\|\begin{pmatrix} A_1 XB_1 \\ A_2 XB_2 \end{pmatrix}- \begin{pmatrix} C_1 \\ C_2 \end{pmatrix}\right\|=\min \] 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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