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In the [[time-invariant]] case, we can define <math> \mathbf{\Phi}</math>, using the [[matrix exponential]], as <math>\mathbf{\Phi}(t, t_0) = e^{\mathbf{A}(t - t_0)}</math>. <ref>{{cite journal |last1=Reyneke |first1=Pieter V. |title=Polynomial Filtering: To any degree on irregularly sampled data |journal=Automatika |date=2012 |volume=53 |issue=4 |pages=382–397|doi=10.7305/automatika.53-4.248 |s2cid=40282943 |url=http://hrcak.srce.hr/file/138435 |doi-access=free |hdl=2263/21017 |hdl-access=free }}</ref>
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]^\mathrm{T}</math>, <math>[0,\ 1,\ \ldots,\ 0]^\mathrm{T}</math>, ..., <math>[0,\ 0,\ \ldots,\ 1]^\mathrm{T}</math>. The corresponding solutions provide the <math>n</math> columns of matrix <math>\mathbf{\Phi}(t, t_0)</math>. Now, from property 4,
<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.
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