# matrix exponential function of a skew hermitian matrix

EDIT: My initial problem was to find the kind of matrices, for which the exponential of is diagonal (and later addition to conditions - unitary), i.e. find all $$A$$, with non-zero entries, such that $$e^A$$ is diagonal (and unitary).

I, mostly experimentally, found that the matrices of the following form $$$$A = \begin{bmatrix} z & k \pi i \\ k \pi i & z \end{bmatrix}, \quad \text{where} \quad z\in \mathbb{C} \quad \text{and} \quad k \in \mathbb{Z}$$$$

give $$$$e^A = \begin{bmatrix} (-1)^ke^z & 0 \\ 0 & (-1)^ke^z \end{bmatrix}.$$$$

Something similar I found for $$A\in \mathbb{C}^{3 \times 3}$$, but these seem to me partial cases, and if they are not, what do you think would be a good approach to prove that? Any directions towards good literature for this or published papers would be very helpful.

After that, must be fairly easy to prove (I presume, taken that the statement is correct) that:

If $$A$$ is skew-Hermitian and $$e^A$$ is diagonal, then $$A$$ can be expressed as

$$$$A = \begin{bmatrix} bi & k \pi i \\ k \pi i & bi \end{bmatrix}, \quad \text{where} \quad b\in \mathbb{R} \quad \text{and} \quad k \in \mathbb{Z}.$$$$

P.S. I am not sure if this is already a proven result or if someone before has worked on this; so would really appreciate if someone who knows could share with that kind of information.

This is not true. Take, for instance, $$A=\left[\begin{smallmatrix}2\pi i&0\\0&0\end{smallmatrix}\right]$$. It is skew-Hermitian and $$e^A=\left[\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right]$$. However, $$A$$ is not of that form that you mentioned.