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We here characterize the minimality of realization of arbitrary lineartime-invariant dynamical systems through (i) intersection of the spectra of therealization matrix and of the corresponding state submatrix and (ii) moving thepoles by applying static output feedback. In passing, we introduce, for a givensquare 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 beequal to the smallest geometric multiplicity among the eigenvalues of A.Finally, we show that the use of a (not necessarily square) realization matrixL to examine minimality of realization, is equivalent to the study of a smallerdimensions, square realization matrix L_sq, which in turn is linked torealization matrices obtained as polynomials in L_sq. Namely a whole family ofsystems.

Minimal state space realization, static output feedback and matrix completion of families of systems