Sylvester equation

Summary

The Sylvester equation is the linear matrix equation $AX + XB = C$, in which the matrices $A$, $B$, $C$ are given and the matrix $X$ is unknown. It has a unique solution exactly when $A$ and $-B$ have no common eigenvalue. It appears in block diagonalisation, in control theory, and as the general form of the Lyapunov equation.