If a matrix A is diagonalizable, then there exists an invertible P so that $A = PDP^{−1}$, where D is a diagonal matrix of eigenvalues of A, and P is a matrix having eigenvectors of A as its columns. In this case, $e^A = Pe^D P^{-1}$.
If A is an $n \times n$ matrix with a single eigenvalue $\lambda$, then there exist a nonnegtive integer $k < n$ such that \(e^{tA} = e^{\lambda t} \left[ \mathbb I + t(A-\lambda \mathbb I)+ \frac{t^2}{2!}(A-\lambda \mathbb I)^2+\cdots+\frac{t^k}{k!}(A-\lambda \mathbb I)^k\right] \,.\)
If $\lambda$ is an eigenvalue of $A$ and $(A-\lambda \mathbb I)^p \, {\mathbf v} = 0$ for some $p\ge 1$, then $\mathbf v$ is called a generalized eigenvector of $A$. When eigenvalues have algebraic multiplicity greater than one, we can compute extra solutions by looking for vectors in the nullspace of $(A-\lambda \mathbb I)^p$.
Ref: The Ordinary Differential Equations Project by Thomas W. Judson.