Abstract: One of the fundamental problems in linear time-invariant multivariable system theory is to obtain minimum realization from a given transfer matrix. In this paper, Rosenbrock's algorithm for minimum realization is improved to reduce the order of the matrix polynomials of the transfer function matrix and to reduce the computing time. As the minimum realization so obtained is not in canonical form, the authors attempt to develop another algorithm to transform the first one into Luenberger's canonical form. Both algorithms are tested on DJS-130 computer.