Full list of eigenvalues of $e^A$ of a matrix $A$

Question

Let $A$ be a square matrix of size $n$ whose eigenvalues are $\lambda_1, ...,\lambda_n$ (counted with algebraic multiplicity).

Then what about the full list of eigenvalues of $e^A$? I suspect that the list is $e^{\lambda_1}, ..., e^{\lambda_n}$ with exactly the same multiplicity as those of $A$. However it seems somewhat tricky to prove it. Could anyone please help me?

Answer 1

Your matrix $A$ is similar (over an algebraically closed extension of your field of scalars) to an upper triangular matrix $T$ and the entries of the main diagonal of $T$ are $\lambda_1,\ldots,\lambda_n$. Besides, the entries of the main diagonal of $e^T$ are $e^{\lambda_1},\ldots,e^{\lambda_n}$. So, …