Abstract: part (Ⅰ) of this work is on the theory of minimal polvnomial matrix and Part (Ⅱ) on the applications of this theory to linear multivariable systems.In Part (Ⅰ), concepts of annihilating polvnomial matrix and the minimal polynomial matrix of a given linear transformation in a vector group are given and the concepts of the generating system and minimal generating system of an invariant subspace for a given linear transformation are given as well. After discussing the basic properties of these concepts the relations between them and the characteristic matrix corresponding to an induced operator of a given linear transformation in any of its invariant subspace are studied in detail. The characteristics of the minimal polynomial matrix for a given vector group and the necessary and sufficient condition for the two generating systems to have the same generating suhspace is given. Using these results we can give the expression for the set of all B’s which makes the system x= Ax+Bu a complete controllable system for a given A.

Key words: pressure tube, brittle fracture, cusp catastrophe, bifurcation set