Diagonalize a Hermitian matrix with a constraint on the unitary transformation. I have a $2n\times 2n$ Hermitian matrix $H$ which I want to diagonalize, with the requirement that the unitary transformation be of the form:
$$
U = \left( \begin{array}{c c}
W & V \\
V^* & W^*
\end{array}
 \right)
$$
in which star stands for complex conjugate and W and V are $n \times n$ matrices. 
Is there a way to do this for a general $H$? If not, what is the restriction on $H$? and how can such $H$ be diagonalized with the above requirement? 
 A: Note first that $U$ is unitary if and only if matrices $W, V$ satisfy the conditions
$$ WW^*+VV^*=I, \quad W^*W+V^*V=I, \quad WV+VW=0. \tag1$$
In the case $n=1$ when $W=\alpha, V=\beta$ are complex numbers we see that either $\beta=0$ and $|\alpha|=1$ or $\alpha=0$ and $|\beta|=1$. Hence
$$ U=\left[ \begin{array}{cc} \alpha & 0\\ 0&\overline{\alpha}\end{array} \right] \quad or
\quad U=\left[ \begin{array}{cc} 0 & \beta\\ \overline{\beta}&0\end{array} \right]. $$
It is clear that a hermitian $2\times 2$ matrix can be diagonalized with such a unitary matrix very rarely.
Assume now that $H$ is a $2n \times 2n$ hermitian matrix which can be diagonalized by a unitary matrix $U$ which has prescribed form. Then
$$ UHU^*=D, $$
where $D$ is a real diagonal matrix. Hence
$$ H=U^*DU.$$
Let
$$ D=\left[ \begin{array}{cc} D_1 & 0\\ 0&D_2\end{array} \right]. $$
Then
$$ H=\left[ \begin{array}{cc} W^* & V\\ V^*&W\end{array} \right]
\left[ \begin{array}{cc} D_1 & 0\\ 0&D_2\end{array} \right]
\left[ \begin{array}{cc} W & V\\ V^*&W^*\end{array} \right]=$$
$$=\left[ \begin{array}{cc} W^*D_1W+VD_2V^* & W^*D_1V+VD_2W^*\\ V^*D_1W+WD_2V^*&V^*D_1+WD_2W^*\end{array} \right]. \tag2$$
Hence one can say that a hermitian matrix can be diagonalized by $U$ of the prescribed form if and only if there exist $n\times n$ matrices $W, V$ satisfying (1) and real diagonal matrices $D_1, D_2$ such that $H$ is of form (2). This characteristion is not very good. One should describe pairs of matrices $W, V$ satisfying (1). It seems to me
that this is actually the problem. I guess that there are not many pairs which satisfy (1) - not many in the sense that one could diagonalize a lot of hermitian matrices with unitaries which are of the above form.
