In nonlinear dynamical system, we have the picard-lindelof existence and uniqueness principle which guarantees existence of unique solution to problem of the type $\dot x = f(x,t), x(0) = x_o$ provided that $f$ is a contraction mapping and $f$ is continuous
We know that this result carries over to the linear and finite dimensional case: (1) $\dot x = Ax, x(0) = x_o$, which has the unique solution $x = \exp(At)x_o$
But how can we use the picard lindelof existence and uniqueness theorem to prove that $x = \exp(At)x_o$ is the unique solution to (1)? We need condition on the continuous and contractiveness of $A$. How do you construct such an $A$ such that it is continuous and contractive? Can someone show how this can be done?
You just have to verify that $x \mapsto Ax$ is Lipschitz, which is not difficult considering the definition of continuity for a linear map. And in finite dimension, all linear maps are continuous.
Then you'll prove that $x = \exp(At)x_o$ is solution of the equation. According to Picard–Lindelöf theorem, the solution is unique.