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:$$ C(A,B) = \left[ B , AB, A^{2}B, A^{3}B,..., A^{n-1}B \right]$$

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.