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How should I solve this matrix equation? What is the solution for $X$?

\begin{equation} BXC +B^TXC^T=D \end{equation}

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up vote 3 down vote accepted

Rewrite it as a system of linear equations $(C^T\otimes B+C\otimes B^T)\operatorname{vec}(X)=\operatorname{vec}(D)$ (see Wikipedia entries on Kronecker product and vectorisation for the meanings of the symbols $\otimes$ and $\operatorname{vec}$) and solve it.

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Nice observation, +1. – 1015 Apr 17 '13 at 4:47

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