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Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation \(A_1X_1B_1+A_2X_2B_2=C\). (English) Zbl 1171.15310

Summary: A matrix \(P\in\mathbb{R}^{n\times n}\) is called a generalized reflection matrix if \(P^T=P\) and \(P^{2}=I\). An \(n\times n\) matrix \(A\) is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix \(P\) if \(A=PAP\) \((A=-PAP)\). In this paper, three iterative algorithms are proposed for solving the linear matrix equation \(A_{1}X_{1}B_{1}+A_{2}X_{2}B_{2}=C\) over reflexive (anti-reflexive) matrices \(X_{1}\) and \(X_{2}\). When this matrix equation is consistent over reflexive (anti-reflexive) matrices, for any reflexive (anti-reflexive) initial iterative matrices, the reflexive (anti-reflexive) solutions can be obtained within finite iterative steps in the absence of roundoff errors. By the proposed iterative algorithms, the least Frobenius norm reflexive (anti-reflexive) solutions can be derived when spacial initial reflexive (anti-reflexive) matrices are chosen. Furthermore, we also obtain the optimal approximation reflexive (anti-reflexive) solutions to the given reflexive (anti-reflexive) matrices in the solution set of the matrix equation. Finally, some numerical examples are presented to support the theoretical results of this paper.

MSC:

15A24 Matrix equations and identities
65F10 Iterative numerical methods for linear systems
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