I am posting this question to learn calculating the matrix power $e^A$ where $A$ is a given matrix in a more beginner-friendly explanation. Definition of this expression is given as $$e^A = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \dots = \sum_{n=0}^\infty \frac{A^n}{n!}$$
What I know:
- If $A$ has $3$ distinct eigenvalues then, we can diagonalize it in the form $A = P\Lambda P^{-1}$ where $\Lambda = \text{diag}(\{\lambda_i: i = 1,2,\dots, n\})$. After this $$e^A = Pe^\Lambda P^{-1}$$ where $e^\Lambda$ is just all elements of $\Lambda$ are exponentiated.
- If $A$ has a "nice" characteristic polynomial and of smaller size then we can benefit from the Cayley Hamilton theorem as in this post.
Main question:
- What to do when the matrix $A$ has $2$ or more same eigenvalues?
- What if there are complex eigenvalues? (this one has been more or less discussed here)
Showing some examples with $3\times 3$ matrices would be greatly appreciated i.e. two cases
- With two same eigenvalues
- With three same eigenvalues
Basic explanations are enough like "If so, use this formula, else this formula", I will try to do the rest of the deeper learning myself.
Thanks in advance.
