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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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