Two theorems regarding the state-space realisation of the transfer-function matrix Z(s)

This letter shows the independence of the matrix products (CAiB, i = 1, 2, ¿) from any state-space realisation of the infinite number of possible realisations1,2 of a transfer-function matrix Z (s). In addition, it shows that, in the sequnce (Z1, Z2, ¿) derived from Z(s), only the first M can be linearly independent, where M is the degree of the minimum polynomial of A.