 Sylvester equation

The Sylvester equation $AX-XB=C$ has a unique solution $X$ for every $C$ exactly when $A$ and $B$ have disjoint spectra.

The Sylvester equation is a type of linear matrix equation that has the general form: \[ AX + XB = C \] where: - \(A\) and \(B\) are given matrices of appropriate dimensions, - \(X\) is the unknown matrix to be solved for, and - \(C\) is a given matrix.