A long division method based on a matrix euclidean algorithm and a continued fraction method based on a matrix Routh algorithm are developed. These are used for constructing a similarity transformation matrix which would transform an observer (controller) block companion form to a controller (observer) block companion form, and for converting a reducible or an irreducible left (right) matrix fraction description to an irreducible right (left) matrix fraction description. The relationships between the state-space models and the corresponding rational matrices of a class of multivariable systems are also explored. The new procedures enable us to enhance the computational aspects of designing self-tuning controllers for on-line adaptive control of a class of multivariable systems.