# A Visualization for Controllability of Linear Systems

This post is an attempt to intuitively explain the controllability matrix rank condition of a linear system. Consider the following linear system:

For the sake of simplicity, we will deal with single input systems. Thus, the matrix B is a column vector. The controllability matrix of the system is calculated as follows:

For controllability, rank(P)=n, where n is the number of state variables. If rank(P)<n, the system is uncontrollable.