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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 $\Bbb R^{m\times n}$ be the set of all $m\times n$ matrices, $S\Bbb R^n$ the set of all symmetric matrices in $\Bbb R^{n\times n}$. For $A\in \Bbb R^{m\times n}$, $\Vert A\Vert$ denotes the Frobenius norm. The authors consider the following two problems. Problem 1. Given $A\in \Bbb R^{m\times n}$, $B\in \Bbb R^{n\times p}$, $C\in \Bbb R^{m\times p}$, find $X\in S\Bbb R^{n}$ such that $AXB=C$. Problem 2. If Problem 1 is consistent, then denote its solutions by ${\cal S}_E$. For given $X_0\in \Bbb R^{n\times n}$, find $\hat{X}\in {\cal S}_E$ such that $$\Vert \hat{X}-X_0\Vert = \min \{\Vert X-X_0\Vert :X\in {\cal S}_E \}.$$ 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 65F10 Iterative methods for linear systems 15A24 Matrix equations and identities
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##### References:
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