Note : $\mathrm{O}(n)$ is $\mathrm{O}(n, \mathbb{R})$.
I tried to solve the following exercise : For every complex orthogonal matrix $g \in \mathrm{O}(n, \mathbb{C})$, i.e. $g^{-1} = {}^\mathrm{t}g$, there exist $k \in \mathrm{O}(n)$ and $X \in M_{n}(\mathbb{R})$ anti-symmetric such that $g = k\exp(\mathrm{i}X)$. From the book Faraut, J. (2018). Analyse sur les groupes de Lie. Calvage & Mounet, there is no solution.
I think this question can be break in these steps :
- ${}^\mathrm{t}\overline{g}g$ is orthogonal hermitian.
- For every $A \in M_n(\mathbb{C})$ orthogonal hermitian, there exist $Y \in M_n(\mathbb{R})$ anti-symmetric such that $A = \exp(\mathrm{i}Y)$.
- Let $Y \in M_n(\mathbb{R})$ anti-symmetric such that ${}^\mathrm{t}\overline{g}g = \exp(\mathrm{i}Y)$. Verify the matrix $g \exp(-\frac{\mathrm{i}Y}{2})$ is real and orthogonal.
1 and 3 is ok for me. For 2, Since $A$ is hermitian, we can write $A = UDU^{*}$ where $U$ is unitary and $D$ is diagonal real matrix, $D = \mathrm{diag}(\lambda_{1}, ... \lambda{n})$. Since $A$ is orthgonal complex, each $\lambda_{i}$ is -1 or 1. So we have $A = \exp(\mathrm{i}U\mathrm{diag}(\delta_{1}, ... \delta_{n})U^{*})$ where $\delta_{i} = \pi$ if $\lambda_{i} = -1$ ; $\delta_{i} = 0$ if $\lambda_{i} = 1$.
How to prove that $U\mathrm{diag}(\delta_{1}, ... \delta_{n})U^{*}$ is real anti-symmetric...? Also, I'm willing to see different solution for the exercise if there is one.