On the solving of matrix equation of Sylvester type

Solutions of two problems related to the matrix equation of Sylvester type is given. In the rst problem, the procedures for linear matrix inequalities are used to construct the solution of this equation. In the second problem, when a matrix is given which is not a solution of this equation, it is required to nd such solution of the original equation, which most accurately approximates the given matrix. For this, an algorithm for constructing a general solution of the Sylvester matrix equation is used. The e ectiveness of the proposed approaches is illustrated on the examples.