It is well known that the necessary and sufficient condition for a pair (A, B) to be completely controllable is that the rank of controllability criterion matrix [B AB…An-1B] equal to n, the dimension of the state vector. Many ways of the proof of the criterion have been contrived by many researchers. Although the point of view that the complete controllability implies the coverage of the whole state space by controllable subspace somehow gives a geometrical interpretation of the controllability criterion, the root of controllability from the geometrical point of view has not yet been disclosed. The author discovered that relative configurations of eigenvectors of A and b (single-input case) play the essential role with regard to controllability. A single-input case is first considered. Realness and multiplicity of eigenvalues are examined in detail. Many examples are provided for a full understanding of the presented theorems. A multi-input case is briefly treated by applying Gopinath's lemma. The problem of dimension of controllable subspace is also examined from the geometrical point of view.
