controllability of LTI systems

Consider the linear time invariant (LTI) system given by:

$$\dot{x}=Ax+Bu$$

where $A$ is an $n\times n$ matrix, $B$ is an $n\times m$ matrix, $u$ is an $m\times 1$ vector - called the control or input vector, $x$ is an $n\times 1$ vector - called the state vector, and $\dot{x}$ denotes the time derivative of $x$.

Definition Of Controllability Matrix For LTI Systems: The controllability matrix of the above LTI system is defined by the pair $(A,B)$ as follows:

$$C(A,B)=[B,AB,A^{2}B,A^{3}B,\mathrm{\dots },A^{n-1}B]$$

Test for Controllability of LTI Systems: The above LTI system $(A,B)$ is controllable if and only if the controllability matrix $C(A,B)$ has rank $n$;i.e. has $n$ linearly independent columns.