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In the [[time-invariant]] case, we can define <math> \mathbf{\Phi}</math>; as <math>\mathbf{\Phi}(t, t_0) = e^{\mathbf{A}(t - t_0)}</math>.
In the [[time-variant]] case, the state-transition matrix <math>\mathbf{\Phi}(t, t_0)</math> can be estimated from the solutions of the differential equation <math>\dot{\mathbf{u}}(t)=\mathbf{A}(t)\mathbf{u}(t)</math> with initial conditions <math>\mathbf{u}(t_0)</math> given by <math>[1,\ 0,\ \ldots,\ 0]^T</math>, <math>[0,\ 1,\ \ldots,\ 0]^T</math>, ..., <math>[0,\ 0,\ \ldots,\ 1]^T</math>. The corresponding solutions provide the <math>n</math> columns of matrix <math>\mathbf{\Phi}(t, t_0)</math>.
<math>\mathbf{\Phi}(t, \tau) = \mathbf{\Phi}(t, t_0)\mathbf{\Phi}(\tau, t_0)^{-1}</math> for all <math>t_0 \leq \tau \leq t</math>. The state-transition matrix must be determined before analysis on the time-varying solution can continue.
== See also ==
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