# How can I solve an orthogonal-constrained Sylvester equation?

I am currently facing a Sylvester equation

$$AX+XB = C$$

where $$A$$, $$B$$, $$C$$ are all symmetric and a special constraint here is that $$X$$ should be orthogonal. The Sylvester equation itself may not hold perfectly and hence the least-square solution is also ok.

• Try a numerical solution first, e.g., $\min_{X^TX=I} \|AX+XB-C\|_F$ using manopt.org Oct 24 '19 at 15:19
• If you relax $X^T X = I$ into $X^T X \preceq I$, you have a semidefinite program. Oct 24 '19 at 18:48
• Thank you for the information. I should have made my expression more clear. That is, I prefer an analytical solution rather than a numerical one. And regarding to relax it into a semidefinite program question, may I say that the solution is still not analytical?
– lisi
Oct 25 '19 at 3:16
• @lisi Analytical solutions are a luxury and a rarity. Oct 25 '19 at 6:10

Let me assume that $$A$$ and $$-B$$ have no common eigenvalues. Without the orthogonality constraint there is then a unique solution $$Y$$ of the Sylvester equation $$AY+YB=C$$, which you can find using known methods. Let $$Y$$ have the singular value decomposition $$Y=U\Sigma V^T$$. Then $$X=UV^T$$ is the orthogonal matrix that minimizes $$\sum_{ij}(X_{ij}-Y_{ij})^2$$. (For a proof, see here.)