I have a skew symmetric matrix $$ C = \begin{pmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \end{pmatrix} $$ and we have the relation $C = U D U^{-1} $, where $D$ is a diagonal matrix. What are the values of $U$ and $D$?
NB: Any values of $U$ and $D$ solve my issue. The $a_i$ values can't be changed.
On
As noted in the comments the three eigen values of $C$ are $$ 0,\quad\mp\, i\, \underbrace{\sqrt{a_1^2 + a_2^2 + a_3^2}}_{\textstyle=:\lambda}\,. $$ The matrix $U$ is $$ \begin{pmatrix}a_1&-a_1 a_3+i\lambda a_2&-a_1 a_3-i\lambda a_2\\ a_2&-a_2 a_3-i\lambda a_1&-a_2 a_3+i\lambda a_1 \\ a_3&a_1^2 + a_2^2 & a_1^2 + a_2^2 \end{pmatrix}\,. $$
Assuming that the $a_i$ are real then $E=iC$ is Hermitean so $E=VF{V ^{-1}}$ with $V$ unitary and $F$ diagonal with real entries (the eigenvalues of $E$). Generically $V$ and $F$ are unique (up to a phase factor in $V$). Now $C=V(-iF)V^{-1}$ so we can take $U=V$ and $D=-iF$. But then $U{e ^D}U ^{-1}$ is not a real matrix. Why do you think it is? And what do you mean with "Any values of $U$ and $D$ solve my issue "?