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An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation \(AXB\)=\(C\). (English) Zbl 1068.65056
Let \(\mathbb R^{m\times n}\) be the set of all \(m\times n\) matrices, \(S\mathbb R^n\) the set of all symmetric matrices in \(\mathbb R^{n\times n}\). For \(A\in \mathbb R^{m\times n}\), \(\| A\| \) denotes the Frobenius norm. The authors consider the following two problems. Problem 1. Given \(A\in \mathbb R^{m\times n}\), \(B\in \mathbb R^{n\times p}\), \(C\in \mathbb R^{m\times p}\), find \(X\in S\mathbb R^{n}\) such that \(AXB=C\). Problem 2. If Problem 1 is consistent, then denote its solutions by \({\mathcal S}_E\). For given \(X_0\in \mathbb R^{n\times n}\), find \(\hat{X}\in {\mathcal S}_E\) such that \[ \| \hat{X}-X_0\| = \min \{\| X-X_0\| :X\in {\mathcal S}_E \}. \] The authors describe an iterative method that determines the solvability of Problem 1 automatically and in the case of solvability computes a solution in an a priori known finite number of steps. Furthermore, the solution to Problem 2 can be found by choosing a suitable initial iteration matrix. It can also be found as the least-norm solution to another equation \(A\bar{X}B=\bar{C}\). The paper is carefully written with detailed and convincing proofs. It also contains a numerical example.

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