The relation between a real matrix and a complex matrix. Let $A$ and $B$ be $n\times n$ real matrices. Show that
$$\det\begin{pmatrix}A & -B \\ B & A \end{pmatrix}=|\det(A+iB)|^{2}.$$

$$\begin{pmatrix}1/2 & -i/2 \\ 1/2 & i/2 \end{pmatrix}\begin{pmatrix}A & -B \\ B & A \end{pmatrix}\begin{pmatrix}1 & 1 \\ i & -i \end{pmatrix}=\begin{pmatrix}A-iB & 0 \\ 0 & A+iB \end{pmatrix}$$
The determinants are $i/2,\det\begin{pmatrix}A & -B \\ B & A \end{pmatrix},-2i$ and $|\det(A+iB)|^2$.
For a holomorphic transformation $\mathbb{C}^n\rightarrow\mathbb{C}^n$, the Jacobian is like $\begin{pmatrix}A & -B \\ B & A \end{pmatrix}$ due to the Cauchy-Riemann equation. On the other hand, if you complexify the (co)tangent bundle and make a coordinate change $dx_i,dy_i\rightarrow dz_i=dx_i+idy_i,d\bar{z}_i=dx_i-idy_i$, the Jacobian will be like $\begin{pmatrix}A-iB & 0 \\ 0 & \overline{A-iB} \end{pmatrix}$. And $\begin{pmatrix}1/2 & -i/2 \\ 1/2 & i/2 \end{pmatrix}$, $\begin{pmatrix}1 & 1 \\ i & -i \end{pmatrix}$ are just the transformation matrices.
