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I am trying to find the least-squares solution $X$ of the following matrix equation

$$AXB+CXD=E$$

Of course, I know that this equation can be written in the form

$$(B^T \otimes A+D^T \otimes C) \operatorname{vec} X = \operatorname{vec} E$$

where $\otimes$ denotes the Kronecker product, and we can take the pseudoinverse of the Kronecker product matrix on the LHS to get $X$. However, I wonder whether we can get the answer in terms of the pseudoinverses $A^{+}$, $B^{+}$, $C^{+}$ and $D^{+}$? I know that if we want to find the least-squares solution of $AXB=E$, we have a solution $X=A^{+}EB^{+}$. Can we write in a similar way a solution of the more complicated case? Thanks!

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  • $\begingroup$ Are the coefficient matrices square? $\endgroup$ – Federico Poloni Oct 15 '17 at 9:24
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This is the Sylvester equation. A simple explicit solution is possible under certain conditions (no common eigenvalues of the matrices $C^{-1}A$ and $-DB^{-1}$), as explained in the Wikipedia page. There are more complicated general methods, see

  1. The explicit solution of the matrix equation $AX−XB=C$ (1995)
  2. Continued-fraction solution of matrix equation $AX-XB=C$(1989)
  3. Explicit solutions of the matrix equation $AX−XB=C$ (1974)
  4. Explicit Solutions of Linear Matrix Equations (1970)
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  • $\begingroup$ Thanks a lot. Because I am studying a low-rank solution to this equation, and it's hard to deal with the rank when it is in vectorization form. I will look at these papers. $\endgroup$ – dave2d Oct 15 '17 at 20:53

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