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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.

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