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

${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t),\;\mathbf {x} (t_{0})=\mathbf {x} _{0}}$ ,

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:^[1][2]

${\displaystyle \mathbf {x} (t)=\mathbf {\Phi } (t,t_{0})\mathbf {x} (t_{0})+\int _{t_{0}}^{t}\mathbf {\Phi } (t,\tau )\mathbf {B} (\tau )\mathbf {u} (\tau )d\tau }$ 

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

${\displaystyle {\begin{aligned}\mathbf {\Phi } (t,\tau )=\mathbf {I} &+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\,d\sigma _{1}\\&+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{\sigma _{1}}\mathbf {A} (\sigma _{2})\,d\sigma _{2}\,d\sigma _{1}\\&+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{\sigma _{1}}\mathbf {A} (\sigma _{2})\int _{\tau }^{\sigma _{2}}\mathbf {A} (\sigma _{3})\,d\sigma _{3}\,d\sigma _{2}\,d\sigma _{1}\\&+\cdots \end{aligned}}}$ 

where ${\displaystyle \mathbf {I} }$  is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.^[2] The series has a formal sum that can be written as

${\displaystyle \mathbf {\Phi } (t,\tau )=\exp {\mathcal {T}}\int _{\tau }^{t}\mathbf {A} (\sigma )\,d\sigma }$ 

where ${\displaystyle {\mathcal {T}}}$  is the time-ordering operator, used to ensure that the repeated product integral is in proper order. The Magnus expansion provides a means for evaluating this product.

The state transition matrix ${\displaystyle \mathbf {\Phi } }$  satisfies the following relationships. These relationships are generic to the product integral.

1. It is continuous and has continuous derivatives.

2, It is never singular; in fact ${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )=\mathbf {\Phi } (\tau ,t)}$  and ${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )\mathbf {\Phi } (t,\tau )=\mathbf {I} }$ , where ${\displaystyle \mathbf {I} }$  is the identity matrix.

3. ${\displaystyle \mathbf {\Phi } (t,t)=\mathbf {I} }$  for all ${\displaystyle t}$  .^[3]

4. ${\displaystyle \mathbf {\Phi } (t_{2},t_{1})\mathbf {\Phi } (t_{1},t_{0})=\mathbf {\Phi } (t_{2},t_{0})}$  for all ${\displaystyle t_{0}\leq t_{1}\leq t_{2}}$ .

5. It satisfies the differential equation ${\displaystyle {\frac {\partial \mathbf {\Phi } (t,t_{0})}{\partial t}}=\mathbf {A} (t)\mathbf {\Phi } (t,t_{0})}$  with initial conditions ${\displaystyle \mathbf {\Phi } (t_{0},t_{0})=\mathbf {I} }$ .

6. The state-transition matrix ${\displaystyle \mathbf {\Phi } (t,\tau )}$ , given by

${\displaystyle \mathbf {\Phi } (t,\tau )\equiv \mathbf {U} (t)\mathbf {U} ^{-1}(\tau )}$ 

where the ${\displaystyle n\times n}$  matrix ${\displaystyle \mathbf {U} (t)}$  is the fundamental solution matrix that satisfies

${\displaystyle {\dot {\mathbf {U} }}(t)=\mathbf {A} (t)\mathbf {U} (t)}$  with initial condition ${\displaystyle \mathbf {U} (t_{0})=\mathbf {I} }$ .

7. Given the state ${\displaystyle \mathbf {x} (\tau )}$  at any time ${\displaystyle \tau }$ , the state at any other time ${\displaystyle t}$  is given by the mapping

${\displaystyle \mathbf {x} (t)=\mathbf {\Phi } (t,\tau )\mathbf {x} (\tau )}$ 

In the time-invariant case, we can define ${\displaystyle \mathbf {\Phi } }$ , using the matrix exponential, as ${\displaystyle \mathbf {\Phi } (t,t_{0})=e^{\mathbf {A} (t-t_{0})}}$ . ^[4]

In the time-variant case, the state-transition matrix ${\displaystyle \mathbf {\Phi } (t,t_{0})}$  can be estimated from the solutions of the differential equation ${\displaystyle {\dot {\mathbf {u} }}(t)=\mathbf {A} (t)\mathbf {u} (t)}$  with initial conditions ${\displaystyle \mathbf {u} (t_{0})}$  given by ${\displaystyle [1,\ 0,\ \ldots ,\ 0]^{\mathrm {T} }}$ , ${\displaystyle [0,\ 1,\ \ldots ,\ 0]^{\mathrm {T} }}$ , ..., ${\displaystyle [0,\ 0,\ \ldots ,\ 1]^{\mathrm {T} }}$ . The corresponding solutions provide the ${\displaystyle n}$  columns of matrix ${\displaystyle \mathbf {\Phi } (t,t_{0})}$ . Now, from property 4, ${\displaystyle \mathbf {\Phi } (t,\tau )=\mathbf {\Phi } (t,t_{0})\mathbf {\Phi } (\tau ,t_{0})^{-1}}$  for all ${\displaystyle t_{0}\leq \tau \leq t}$ . The state-transition matrix must be determined before analysis on the time-varying solution can continue.

• Magnus expansion
• Liouville's formula

• Baake, M.; Schlaegel, U. (2011). "The Peano Baker Series". Proceedings of the Steklov Institute of Mathematics. 275: 155–159. doi:10.1134/S0081543811080098. S2CID 119133539.
• Brogan, W.L. (1991). Modern Control Theory. Prentice Hall. ISBN 0-13-589763-7.