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!


Check out the work of Beckermann and Townsend.

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

  • $\begingroup$ 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
    Nov 14 '17 at 16:32
  • $\begingroup$ @XavierXiao You have changed the question. $\endgroup$
    – Igor Rivin
    Nov 14 '17 at 16:37

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