On minimum spanning blocks in discrete linear systems

The question of spanning blocks in linear discrete systems arises in many adaptive identification and control problems and is related to the convergence of these algorithms. Specifically, in block-invariant adaptive control, it is of importance to know the minimum length of these spanning blocks. Establishing this minimum length is the topic of the present note. It is found to be equal to the sum of the dimensions of the system´s state space and its controllable subspace.