Let (A+i B) be a full-rank Hermitian matrix of order (n). Show that the inverse matrix (C+i

Question:

Let $A+i B$ be a full-rank Hermitian matrix of order $n$. Show that the inverse matrix $C+i D$ is also Hermitian and satisfies the pair of equations

\[
A D+B C=0 ; \quad A C-B D=I_{n} .
\]

Deduce that the $2 n \times 2 n$ real symmetric matrices

\[
\left(\begin{array}{rr}
A & B \\
-B & A
\end{array}ight) \quad \text { and } \quad\left(\begin{array}{rr}
C & D \\
-D & C
\end{array}ight)
\]
are mutual inverses. What does this matrix isomorphism imply about the relation between complex Gaussian vectors and real Gaussian vectors?