I need to get the jacobian of the function $x(t) = e^{At} x_0 $. so, I was thought about applying the vectorization and Kronecker product:
$ d \, vec \, x = (x_0^T \otimes I) \, d \, vec (e^At)$.
But how should I continue?... I don't know how to write $(e^At)$ in terms of A
A is diagonalizable, and T is an orthonormal set of eigenvectors.
$A = T diag [\lambda_1,..., \lambda_n ] T^{-1}$, so
$e^{At} = T diag [e^\lambda_1,..., e^\lambda_n ] T^{-1}$