Solving the Sylvester Equation AX-XB=C when $\sigma(A)\cap\sigma(B)\neq\emptyset$

Abstract

The method for solving the Sylvester equation $AX-XB=C$ in complex matrix case, when $\sigma(A)\cap\sigma(B)\neq \emptyset$, by using Jordan normal form is given. Also, the approach via Schur decomposition is presented.