A complotely new algorithm for calculating a state space minimal realization of a rational transfer function matrix is given. Tho algorithm consists of four major steps : calculation of the least common denominators of the columns of the transfer function matrix ; calculation of the greatest common right divisor of two polynomial matrices ; factoring the divisor from the two matrices ; and expansion of the resulting relatively right prime matrices into constant matrices. The transfer function matrix need not be proper and the result of the calculations leaves the polynomial part of the transfer function matrix unaltered, thus allowing a useful check on the accuracy of the computations to be incorporated. The results of several examples show that this new algorithm is superior in terms of computing time to other methods and is very accurate.