Linear dynamical systems are dynamical systems whose evolution functions are linear. While dynamical systems, in general, do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point.

In a linear dynamical system, the variation of a state vector (an ${\displaystyle N}$-dimensional vector denoted ${\displaystyle \mathbf {x} }$) equals a constant matrix (denoted ${\displaystyle \mathbf {A} }$) multiplied by ${\displaystyle \mathbf {x} }$. This variation can take two forms: either as a flow, in which ${\displaystyle \mathbf {x} }$ varies continuously with time

or as a mapping, in which ${\displaystyle \mathbf {x} }$ varies in discrete steps

These equations are linear in the following sense: if ${\displaystyle \mathbf {x} (t)}$ and ${\displaystyle \mathbf {y} (t)}$ are two valid solutions, then so is any linear combination of the two solutions, e.g., ${\displaystyle \mathbf {z} (t)\ {\stackrel {\mathrm {def} }{=}}\ \alpha \mathbf {x} (t)+\beta \mathbf {y} (t)}$ where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are any two scalars. The matrix ${\displaystyle \mathbf {A} }$ need not be symmetric.

Linear dynamical systems can be solved exactly, in contrast to most nonlinear ones. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its fixed points. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems.

If the initial vector ${\displaystyle \mathbf {x} _{0}\ {\stackrel {\mathrm {def} }{=}}\ \mathbf {x} (t=0)}$ is aligned with a right eigenvector ${\displaystyle \mathbf {r} _{k}}$ of the matrix ${\displaystyle \mathbf {A} }$, the dynamics are simple

where ${\displaystyle \lambda _{k}}$ is the corresponding eigenvalue; the solution of this equation is

as may be confirmed by substitution.