  We here characterize the minimality of realization of arbitrary linear
time-invariant dynamical systems through (i) intersection of the spectra of the
realization matrix and of the corresponding state submatrix and (ii) moving the
poles by applying static output feedback. In passing, we introduce, for a given
square matrix A, a parameterization of all matrices B for which the pairs (A,
B) are controllable. In particular, the minimal rank of such B turns to be
equal to the smallest geometric multiplicity among the eigenvalues of A.
Finally, we show that the use of a (not necessarily square) realization matrix
L to examine minimality of realization, is equivalent to the study of a smaller
dimensions, square realization matrix L_sq, which in turn is linked to
realization matrices obtained as polynomials in L_sq. Namely a whole family of
systems.
