7. If the system is [[time-invariant]], we can define <math> \mathbf{\Phi}</math>; as:
: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, therethe arestate-transition manymatrix differentcan functionsbe thatestimated mayfrom satisfythe thesesolutions requirementsof the differential equation <math>\mathbf{x}(t)=\mathbf{\Phi}(t, and\tau)\mathbf{x}(\tau)</math> thewith solutioninitial isconditions dependent<math>[1,\ on0,\ \ldots,\ 0]^T</math>, <math>[0,\ 1,\ \ldots,\ 0]^T</math>, ..., <math>[0,\ 0,\ \ldots,\ 1]^T</math>. The corresponding solutions provide the structure<math>n</math> columns of thematrix <math>\mathbf{\Phi}(t, systemt_0)</math>. The state-transition matrix must be determined before analysis on the time-varying solution can continue.
