Autonomous Continuous-Time Linear Dynamical Systems
Definition
An autonomous continuous-time
linear dynamical system is a matrix $A
\in \R ^{n \times n}$.
It models the behavior of a
signal $x: \R \to \R ^n$
by
\begin{equation} \dot{x} = Ax, \label{defining} \end{equation}
where $\dot{x}$ is notation for $\frac{d}{dt}
x(t)$.
$A$ is called the dynamics
matrix.
A signal $x$ satisfying Equation~\eqref{defining} is called a solution or a trajectory. For $t \in \R $, $x(t) \in \R ^n$ is called the state and $\R ^n$ is called the state space. $n$ is called the state dimension.