I am trying to compute the matrix exponential $e^{At}$ for the matrix $A=\begin{pmatrix} 0 &0 \\ 1&0 \end{pmatrix}$. In this case, I have computed the eigenvalues, which are in $\lambda=0$ with algebraic multiplicity 2, and the generalized eigenvectors are $u_1=\begin{pmatrix} 0\\ 1 \end{pmatrix}$ and $u_2=\begin{pmatrix} 1\\ 0 \end{pmatrix}$.
how do I proceed now for computing the matrix exponential?
The solution for this precise $A$ is written in the above remarks.
More general, if you know that the set of all generalized eigenvectors still spans the whole space, you can still "pseudo-diagonalize" such a matrix. Hereby, by transforming into a suitable basis, you don't get rid of all off-diagonal entries, but still you can explicitly compute the exponentials. Maybe you want to look up
https://en.wikipedia.org/wiki/Jordan_normal_form