I have a real-valued, antisymmetric matrix (A and B real):
$$
H=\begin{bmatrix}
0 & A & 0 \\
-A & 0 & B \\
0 & -B & 0 \\
\end{bmatrix}
$$
which can be diagonalized with a unitary transformation:
$$
S= \frac{1}{\sqrt{A^2+B^2}} \begin{bmatrix}
B & \frac{-A}{\sqrt{2}} & \frac{-A}{\sqrt{2}} \\
0 & \frac{-i}{\sqrt{2}}\sqrt{A^2+B^2} & \frac{i}{\sqrt{2}}\sqrt{A^2+B^2} \\
A & \frac{B}{\sqrt{2}} & \frac{B}{\sqrt{2}} \\
\end{bmatrix}
$$
such that
$$
D=S^\dagger H S = \begin{bmatrix}
0 & 0 & 0 \\
0 & i\lambda & 0 \\
0 & 0 & -i\lambda \\
\end{bmatrix}
$$
where $\lambda=\sqrt{A^2+B^2}$
It is a known property of skew symmetric matrices that they can be brought to the block diagonal form containing the same eigenvalues as follows:
$$
\Sigma=Q^T H Q = \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & \lambda \\
0 & -\lambda & 0 \\
\end{bmatrix}
$$
where $Q$ is a real orthogonal matrix. How do you generally construct the matrix $Q$? I have found two unitary transformations
$$
W_{1} = \begin{bmatrix}
1 & 0 & 0 \\
0 & \frac{-i}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
0 & \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} \\
\end{bmatrix}
$$
and
$$
W_{2} = \begin{bmatrix}
1 & 0 & 0 \\
0 & \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} \\
0 & \frac{i}{\sqrt{2}} & \frac{-1}{\sqrt{2}} \\
\end{bmatrix}
$$
which transform $D$ to $\Sigma$ as $\Sigma=W_{j}DW_{j}^\dagger$ for $j=1,2$. Then we have $H=TDT^\dagger$ where $T=SW_{j}^\dagger$. I would expect $T$ to be the real orthogonal matrix $Q$ I am looking for (or its transpose), but instead it is a complex-valued matrix (unitary by construction). How do I construct the real orthogonal matrix $Q$ in this example (and in general)?
Similar questions on stackexchange that I couldn't figure out the answer from are:
Best Answer
In general, if $u\pm iv$ are eigenvectors to a conjugate pair of eigenvalues $a\pm ib$ of a real normal matrix $H$, we must have $\|u\|=\|v\|$ and $u\perp v$. Therefore, with respect to the ordered basis $\{u,v\}$, the matrix representation of the linear map $x\mapsto Mx$ when restricted on the linear span of $u,v$ is given by $\pmatrix{a&-b\\ b&a}$.
In your case, the conjugate pair of eigenvectors are $u\pm iv$, where $u=\frac1{\sqrt{2(A^2+B^2)}}(-A,0,B)^\top$ and $v=\frac1{\sqrt{2(A^2+B^2)}}(0,\sqrt{A^2+B^2},0)^\top$. The matrix $H$ also has an eigenvector $w=\frac1{A^2+B^2}(B,0,A)^\top$ in its null space. Therefore, if you take $Q=\pmatrix{\frac{w}{\|w\|},\frac{u}{\|u\|},\frac{v}{\|v\|}}=\pmatrix{w,\sqrt{2}u,\sqrt{2}v}$, then $Q$ is real orthogonal and $Q^THQ$ will give you the desired block-diagonal form $\Sigma$.