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In control theory, the **state-transition matrix** is a matrix whose product with the state vector at an initial time gives at a later time . The state-transition matrix can be used to obtain the general solution of linear dynamical systems.

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 are the states of the system, is the input signal, and are matrix functions, and is the initial condition at . Using the state-transition matrix , the solution is given by:^{[1]}^{[2]}

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 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

where 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 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 and , where is the identity matrix.

3. for all .^{[3]}

4. for all .

5. It satisfies the differential equation with initial conditions .

6. The state-transition matrix , given by

where the matrix is the fundamental solution matrix that satisfies

- with initial condition .

7. Given the state at any time , the state at any other time is given by the mapping

In the time-invariant case, we can define , using the matrix exponential, as . ^{[4]}

In the time-variant case, the state-transition matrix can be estimated from the solutions of the differential equation with initial conditions given by , , ..., . The corresponding solutions provide the columns of matrix . Now, from property 4, for all . The state-transition matrix must be determined before analysis on the time-varying solution can continue.