In control theory and dynamical systems theory, the state-transition matrix is a matrix function that describes how the state of a linear system changes over time. Essentially, if the system's state is known at an initial time ${\displaystyle t_{0}}$, the state-transition matrix allows for the calculation of the state at any future time ${\displaystyle t}$.

The matrix is used to find the general solution to the homogeneous linear differential equation ${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)}$ and is also a key component in finding the full solution for the non-homogeneous (input-driven) case.

For linear time-invariant (LTI) systems, where the matrix ${\displaystyle \mathbf {A} }$ is constant, the state-transition matrix is the matrix exponential ${\displaystyle e^{\mathbf {A} (t-t_{0})}}$. In the more complex time-variant case, where ${\displaystyle \mathbf {A} (t)}$ can change over time, there is no simple formula, and the matrix is typically found by calculating the Peano–Baker series.

The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form

where ${\displaystyle \mathbf {x} (t)}$ are the states of the system, ${\displaystyle \mathbf {u} (t)}$ is the input signal, ${\displaystyle \mathbf {A} (t)}$ and ${\displaystyle \mathbf {B} (t)}$ are matrix functions, and ${\displaystyle \mathbf {x} _{0}}$ is the initial condition at ${\displaystyle t_{0}}$. Using the state-transition matrix ${\displaystyle \mathbf {\Phi } (t,\tau )}$, the solution is given by:

The first term is known as the zero-input response and represents how the system's state would evolve in the absence of any input. The second term is known as the zero-state response and defines how the inputs impact the system.

The most general transition matrix is given by a product integral, referred to as the Peano–Baker series

where ${\displaystyle \mathbf {I} }$ is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique. The series has a formal sum that can be written as