Solving matrix polynomial equations

Matrix equations and systems of matrix equations are widely used in problems of optimization of control systems, in mathematical economics. However, methods for solving them are developed only for the most popular matrix equations – the Riccati and Lyapunov equations, and there is no universal approach to solving problems of this class. This paper discusses methods for solving matrix polynomial equations of arbitrary order with matrix and vector unknowns. An approach to calculating tuples of solutions of polynomial matrix equations, which is based on the theory of branched chain fractions, is given. It should be noted that we are talking not only about numerical but also symbolic methods of solution. The paper also presents a computational scheme for systems of second-degree polynomial matrix equations with many unknowns. The solution is developed into a continued matrix fraction. Sufficient signs of convergence of branched continued matrix fractions to the solutions and the criteria for completing calculations in iterative procedures are formulated. The results of numerical experiments are presented, confirming the validity of the theoretical calculations and the effectiveness of the proposed methods. Keywords: matrix polynomial equations, branched continued fractions with matrix elements, convergence to solution.