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A Lyapunov matrix equation can be converted, by using the Jordandecomposition theorem for matrices, into an equivalent Lyapunov matrix equationwhere the matrix is a Jordan matrix. The Lyapunov matrix equation with Jordanmatrix can be reduced to a system of Sylvester-Lyapunov type matrix equations.We completely solve the Sylvester-Lyapunov type matrix equations correspondingto the Jordan block matrices of the initial matrix.

Lyapunov matrix equation $A^HX+XA+C=O_r$ with $A$ a Jordan matrix