Determinants of block matrices Let $A,B \in \mathbb{R}^{n,n}$.
Now $C = \begin{pmatrix}  A & iB \\ -iB & A \end{pmatrix}$ and $D = \begin{pmatrix} A & B \\ -B & A \end{pmatrix}$.
Show that $\det(C) \in \mathbb{R}$ and $\det(D) \ge 0$.
I tried to transform $C$ so I could use $\det \begin{pmatrix} E & F \\ 0 & G \end{pmatrix} = \det(E)\det(G)$, but I didn't manage to. I have no clue how to show $\det(D) \ge 0$.
I'd rather have hints than fully-fledged solutions. Thanks.
 A: Here's a hint for (1): think about scaling the first $n$ rows by $-i$. 
Still thinking about (2). I haven't worked this out, but I think considering the signed permutations way of calculating the determinant, and pairing $\sigma$ with $\sigma^{-1}$ might be helpful. 
edit: Yeah, okay, just got back, have something kinda similar to levap. A matrix of the form $D$ is exactly a complex linear one, under the identification $\mathbb{R}^{2n} \rightarrow \mathbb{C}^n$, where $\mathbb{R}^{2n}$ has ordered basis $x_1, \ldots, x_n, y_1, \ldots, y_n$ and $\mathbb{C}^n$ $z_i = x_i + i y_i$. 
With this, one knows as an operator on $\mathbb{C}^n$ one can write it in Jordan canonical form. If one has a Jordan block: $$\begin{pmatrix} \lambda & - & \ldots \\ & \lambda & \ldots \end{pmatrix}$$with basis $w_i$, setting $w_i = x'_i + iy'_i$, in the ordered basis $x_1', y_1', x_2', y_2' \ldots$, it's easy to check that $D$ has determinant $|\lambda|^r$, where $r$ is the size of the Jordan block.  
A: *

*Consider $\begin{pmatrix}I\\&zI\end{pmatrix}C\begin{pmatrix}I\\&\bar{z}I\end{pmatrix}$ for some complex numbers $z$ of modulus 1.

*As invertible matrices are dense in the matrix space and determinant is a continuous function in matrix entries, we may assume that $A$ is invertible. Using the block determinant formula, we get $\det D = \det A \det(A+BA^{-1}B) = (\det A)^2\det(I+A^{-1}BA^{-1}B)$. Now for any real matrix $X$, argue why $\det(I+X^2)$ must be nonnegative.

A: I'll answer the question about $\det(D)$. Define the matrix
$$ J_0 = \left( \begin{array}{cc} 0 & -I \\ I & 0 \end{array} \right) \in \mathrm{GL}_{2n}(\mathbb{R})$$
and note that $J_0^2 = -I$ and that the matrix $J_0$ commutes with your matrix $D$. Now consider all matrices (and maps) as matrices over the complex numbers (on $\mathbb{C}^{2n}$). The matrix $J_0$ is diagonalizable over the complex number with eigenvalues $\pm i$. Denote the eigenspaces by
$$ V^{\pm} = \mathrm{ker}(J \pm iI) \subset \mathbb{C}^{2n}. $$
Each eigenspace is of complex dimension $n$ and the antilinear conjugation map $S : \mathbb{C}^{2n} \rightarrow \mathbb{C}^{2n}$ given by
$$ S(z_1, \ldots, z_{2n}) = (\bar{z}_1, \ldots, \bar{z}_{2n})$$
exchanges between the eigenspaces $V^{+}$ and $V^{-}$. Since $J_0$ and $D$ commute, the spaces $V^{\pm}$ are $n$ dimensional complex invariant spaces of $D$. Considering $D$ as a linear map over $\mathbb{C}^{2n}$, we have 
$$ \det(D) = \det(D_+) \det(D_{-}) = d_{+} d_{-} $$
where the maps $D_{\pm} : V^{\pm} \rightarrow V^{\pm}$ are just the restrictions of $D$ to the invariant subspaces $V^{\pm}$. Let us show that $\bar{d}_{+} = d_{-}$ and so $\det(D) = d_{+} \bar{d}_{+} = |d_{+}|^2 \geq 0$. Because $D$ is a real matrix, it commutes as a linear map with the complex conjugation $S$. This means we have the diagram
$$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!}
\newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.}
%
\begin{array}{llllllllllll}
V^{+} & \ra{D_{+}} & V^{+} \\
\da{S} & & \da{S} \\
V^{-} & \ra{D_{-}} & V^{-} \\
\end{array} $$
where $S$ is an antilinear isomorphism. If $S$ would be a linear isomorphism, it would imply that $\det(D_{+}) = \det(D_{-})$. Since $S$ is antilinear, it implies that the determinants must be conjugate.

This might look highly unmotivated if you are unfamiliar with the notion of a complex linear structure on a vector space. Under the identification of $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ given by 
$$ (z_1, ..., z_n) = (x_1 + iy_1, ..., x_n + iy_n) \to (x_1, ..., x_n, y_1, ..., y_n) $$
the complex matrices $A + iB \in \mathrm{GL}_n(\mathbb{C})$ are identified with the matrices of the form $D$ in $\mathrm{GL}_{2n}(\mathbb{R})$. This matrices are complex linear and so commute with "multiplication by $i$", which corresponds under the identification to multiplication by $J_0$.
