A characterization of eAtand a constructive proof of the controllability criterion

A differential equation characterizing the functions
\alpha_{i}(t)
, which arise when e
^At
is expressed as
\alpha_{0}(t)I + ... + \alpha_{n-1}(t) A^{n-1}
, is derived. It is shown that the set of functions
\{\alpha_{i}(t)}\
is linearly independent over any nonzero interval. Using this fact, a constructive proof is given for the well-known criterion for a linear time-invariant system to be controllable, namely, rank
[B|AB| ... |A^{n-1}B] = n
.