On the structure of the bases of all possible controllability subspaces  of a controllable pair [ A, B] in canonical form

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Abstract
The problem of determining the structure of the basis matrices of all possible controllability subspaces  of a controllable pair [Ã, [Btilde]] in the Brunovski (1966) and Luenberger (1967) controllable canonical form is considered. Departing from a characterization of the c.s.'s of [Ã, [Btilde]] given by Warren and Eckberg (1975) it is shown that to every pair A, B in the Brunovski (1966) and Luenberger (1967) controllable canonical form, there corresponds a unique polynomial matrix X(8) which has a canonical structure. Using the results on rational vector spaces obtained by Forney (1975) it is seen that this polynomial matrix qualifies as a minimal basis which uniquely identifies a rational vector space (s). A correspondence between the polynomial n-tuples x(8)∊(8) and the c.s.'s  of [Ã, [Btilde]] loads to simple expressions that describe the structure of the bases of all c.s.  of [Ã, [Btilde]] of all possible dimensions.