In this paper, an explicit solution to right coprime factorization of transfer function based on Krylov matrix and Pseudo-Controllability Indices is investigated. The proposed approach only needs to solve a series of linear equations. Based on these solutions to right coprime factorization, a complete, analytical, and explicit solution to the generalized Sylvester matrix equation AV - VF = BW with F being an arbitrary known matrix with arbitrary eigenvalues, is proposed. The primary feature of this solution is that the matrix F does not need to be transformed into any canonical form. The results provide great convenience to the computation and analysis of the solution to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.
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