When $A$ is diagonalisable, $\vec{x}_{k+1}=A\vec{x}_k$ implies that $\vec{x}_k = c_1\lambda_1^k\vec{v}_1 +...+c_n\lambda_n^k\vec{v}_n$ because an eigenbasis exists and any $\vec{x}$ can be decomposed into a linear combination of $A$'s eigenvectors.
But when $A$ is not diagonalisable, is there a related general formula for (or a general technique to find) $\vec{x}_k$ in terms of $A$'s eigenvalues and eigenvectors?
I ask this because I have just learnt that for the differential system $\dot{\vec{x}}=A\vec{x}$, a fundamental solution set always exists so that even when $A$ is defective, $\vec{x}$ can still be expressed in terms of $A$'s eigenvectors and eigenvalues.