The controllability matrix for the constant-coefficient system of ordinary differential equations 
, with 
state variables and 
inputs, where 
and 
are positive integers,
is the 
block matrix
![C=[B AB A^2B ... A^(n-1)B].](/images/equations/ControllabilityMatrix/NumberedEquation1.svg) |
The column space of 
consists of the states reachable from the zero state. The
system is controllable, meaning that any initial state can be driven to any final
state by a suitable input over a finite time interval,
iff 
has matrix rank 
(Massachusetts Institute of Technology 2010, pp. 10-1-10-2).
For example, ![A=[0 1; 0 0]](/images/equations/ControllabilityMatrix/Inline10.svg)
and ![B=[0; 1]](/images/equations/ControllabilityMatrix/Inline11.svg)
give
![C=[0 1; 1 0],](/images/equations/ControllabilityMatrix/NumberedEquation2.svg) |
which has matrix rank 2, so the system is controllable.
The same block matrix is also called a reachability matrix in control theory. This usage is distinct from the reachability matrix of a directed graph.
The Wolfram Language computes the controllability matrix of such a state-space model using ControllabilityMatrix[ssm].
