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This paper is concerned with minimal factorizations of rational matrix functions. The treatment is based on a new geometrical principle. In fact, it is shown that there is a one-to-one correspondence between minimal factorizations on the one hand and certain projections on the other. Considerable attention is given to the problem of stability of a minimal factorization. Also the numerical aspects are discussed. Along the way, a stability theorem for solutions of the matrix Riccati equation is obtained.Introduction. The problem of factorizing a rational matrix-valued function W(A) into "simpler" rational factors has network theory as one of its origins. In this theory W(A) appears as a transfer function of a network. Its minimal factorizations (see Chapter II) are of particular interest because it allows one to obtain the network by a cascade connection of elementary sections which have the simplest synthesis [6], [22]. In the present paper the treatment of the factorization problem is based on a new geometrical principle. This principle has been observed independently by the first three authors and by the fourth (and has been communicated at a miniconference on Operators and System Theory held at Amsterdam and Delft, February, 1978). For the fourth author network theory [22], [23] has been the main motivation, while the first three authors were inspired by [31, [7], [20].The new geometrical principle referred to allows for a unifying approach to seemingly disjoint topics such as the network problems mentioned above, the matrix Riccati equation [19], the factorization theory of characteristic functions for linear operators [7], the theory of Wiener-Hopf (or spectral) factorization [10], [11] and the divisibility theory of operator polynomials [3], [12], [13]. Here we treat only the first two topics; the other connections will be investigated in detail in a forthcoming publication [5]. The problem of computing numerically the minimal factors of a transfer function led us to investigate the stability of divisors under small perturbations. We pay considerable attention to the measure of stability. The matrix functions studied here are viewed as transfer functions of systems. A system consists of three matrices A, B, and C, of appropriate sizes, and the corresponding transfer functions are of the form W(A)=I+C(AI-A)-IB,where A is the complex variable and 1 the identity matrix. In the first chapter multiplication and division of transfer functions are described in terms of systems. Applications to matrix Riccati equations are also considered here. The special type of minimal factorization and its properties are studied in Chapter II. In geometrical terms an explicit description of all minimal factors is given. Stability and numerical aspects are studied in the last two chapters. Throughout the paper we confine ourselves to the finite dimensional case, but with minor modifications the results of Chapters I and III are also valid in the infinite dimensional situation (see [5]).