A definition of centralized fixed modes of a system, which is an adoption of the definition of fixed modes for a decentralized system [1], is made. It is shown that the centralized fixed modes of a system can be easily and efficiently calculated in terms of the eigenvalues of the system, and that the calculation of such fixed modes enables one to determine, in a numerically efficient way, whether a system is controllable, observable, stabilizable, detectable. An efficient algorithm for reducing a system to minimal realization form is then given. The algorithms have been effectively used on systems of order up to 100.