= Controllability matrix
{title2=$\mathcal C=\begin{bmatrix}B&AB&\cdots&A^{n-1}B\end{bmatrix}$}
For the <linear time-invariant system> $\dot x=Ax+Bu$, with $A\in\mathbb R^{n\times n}$ and $B\in\mathbb R^{n\times m}$, the <controllability matrix> has $n$ rows and $nm$ columns. The columns of $B$ are directions influenced directly by the <control inputs>; $AB,A^2B,\ldots$ account for how the dynamics carry their influence into further directions. The <column space> of the <controllability matrix> is the zero-state <reachable set> for any $T>0$ when inputs are unrestricted. https://underactuated.csail.mit.edu/acrobot.html#controllability[Russ Tedrake, Underactuated Robotics: Controllability of linear systems]{ref}
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