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For the Sylvester matrix equation $AX+XB=C$, I want to find the least-squares solution $X$ via

$$\begin{array}{ll} \text{minimize} & \| AX + XB - C \|_{\text{F}}^2\\ \text{subject to} & \mbox{rank} (X) \leq k\end{array}$$

I searched broadly online but couldn't find any literature on this. When dealing with a rank constraint, I think that we will use the Eckart-Young Theorem somewhere. However, if we solve the Sylvester equation without the rank constraint using the Kronecker product, it's hard to deal with the rank of $X$ when $X$ is in vectorized form. I also tried to do SVD on $A$ and $B$ but could not proceed either.

Does anyone have any idea about this problem? Or is there literature on this already? Thanks!

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Check out the work of Beckermann and Townsend.

Beckermann, Bernhard; Townsend, Alex, On the singular values of matrices with displacement structure, ZBL06803120.

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  • $\begingroup$ Thanks, I have read the main theorem, and I think here I want the least square solution, and therefore I don't need the Sylvester equation to hold exactly. Also the rank here is exact, not numerical rank. $\endgroup$
    – dave2d
    Commented Nov 14, 2017 at 16:32
  • $\begingroup$ @XavierXiao You have changed the question. $\endgroup$
    – Igor Rivin
    Commented Nov 14, 2017 at 16:37

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