The state-transition matrix must be determined before analysis on the time-varying solution can continue.

The state-transition matrix can be used to obtain the general solution of linear dynamical systems.

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 .

If we have an input , an output , and a weighting pattern then a realization is any triple of matrices such that where is the state-transition matrix associated with the realization.