Solutions to a family of matrix equations by using the Kronecker matrix polynomials

Closed form solutions to a family of generalized Sylvester matrix equation in form of ∑_{i = 0}^φ{symbol} A_i XFⁱ + ∑_{i = 0}^ψ B_k YF^k = ∑_{j = 0}^φ E_j RF^j are given by using the so-called Kronecker matrix polynomials. It is found that the structure of the solutions is independent of the orders φ{symbol}, ψ and φ. This type of uniform closed form solutions includes our early results as special cases. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in linear systems.