Let the three matrices $\sum (N) = (G(N),F(N)H(N))$ define a linear constant system of least degree which realizes the set of numbers $f_1 , \cdots ,f_N $ regarded as a partial impulse response of a system. An algorithm has been developed for recursively calculating the minimal partial realizations for each $N = 1,2, \cdots $ such that \[ \cdots \sum {(N - k)} \subseteq \sum {(N) \subseteq \cdots .} \] This algorithm differs from the previous ones, such as that of B. L. Ho’s, in that there is a recursion on N as well. Because of this, no a priori guess of the order of the system is required. Moreover, an addition of terms to the initial sequence causes the computation of only a few new elements. When combined with another algorithm for factoring covariance matrices the given algorithm permits a recursive identification of linear random systems. No earlier recursive identification methods seem to appear in the literature. Finally, a categorical description of the abstract realizations is given.