<s>[INST] Find two independent orthogonal eigenvectors We are given that $C$ is an $n\times n$ complex matrix, and $C$ is Hermitian ($C = C^*$). Now we define $C = A + Bi$ where $A$ and $B$ are real matrices.
Let $M$ be defined by
\begin{bmatrix}
    A & -B \\
    B & A
\end{bmatrix}
Given that $\lambda$ is an eigenvalue of $C$ with eigenvector $z = x + iy$, we want to find two independent orthogonal eigenvectors of $M$ with eigenvalue $\lambda$. 
I found that 
\begin{bmatrix}
     z \\
     -iz
\end{bmatrix}
is one eigenvector. How do I find the other one?
 [/INST] Given a Hermitian matrix
$C = C^\dagger \in M_{n \times n}(\Bbb C), \tag 1$
which has an eigenvalue $\lambda$ with associated eigenvector $z$:
$Cz = \lambda z, \tag 2$
it is well-known that
$C^\dagger = C \Longrightarrow \lambda \in \Bbb R; \tag 3$
since the entries of $C$ lie in $\Bbb C$, with real $\lambda$, we will generally have
$z \in \Bbb C^n; \tag 4$
that is,
$z = x + iy, \tag 5$
with
$x, y \in \Bbb R^n; \tag 6$
thus we may write
$C(x + iy) = \lambda(x + iy); \tag 7$
now (1) implies we also have
$C = A + iB, \; A, B \in M_{n \times n}(\Bbb R); \tag 8$
thus (7) becomes
$(A + iB)(x + iy) = \lambda x + i \lambda y, \tag 9$
or
$(Ax - By) + i(Ay + Bx) = \lambda x + i \lambda y; \tag{10}$
equating the real and imaginary parts of either side yields
$Ax - By = \lambda x, \tag{11}$
$Ay + Bx = \lambda y. \tag{12}$
We can now in fact find two real orthogonal eigenvectors for
$M = \begin{bmatrix} A & -B \\ B & A \end{bmatrix}, \tag{13}$
each corresponding to the eigenvalue $\lambda$; set
$w = \begin{pmatrix} x \\ y \end{pmatrix}; \tag{14}$
$Mw = \begin{bmatrix} A & -B \\ B & A \end{bmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} Ax - By \\ Bx + Ay \end{pmatrix} = \begin{pmatrix} \lambda x \\ \lambda y \end{pmatrix} = \lambda \begin{pmatrix} x \\ y \end{pmatrix} = \lambda w; \tag{15}$
now set
$v = \begin{pmatrix} -y \\ x \end{pmatrix}; \tag{16}$
$Mv = \begin{bmatrix} A & -B \\ B & A \end{bmatrix} \begin{pmatrix} -y \\ x \end{pmatrix} = \begin{pmatrix} -Ay - Bx \\ -By + Ax \end{pmatrix} = \begin{pmatrix} -\lambda y \\ \lambda x \end{pmatrix} = \lambda \begin{pmatrix} -y \\ x \end{pmatrix} = \lambda v; \tag{17}$
furthermore,
$\langle v, w \rangle = \begin{pmatrix} -y \\ x \end{pmatrix}^T \begin{pmatrix} x \\ y \end{pmatrix} =  \begin{pmatrix} -y^T & x^T \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = -y^Tx + x^Ty; \tag{18}$
$y^T x, x^Ty \in \Bbb R \Longrightarrow y^Tx = (y^Tx)^T = x^Ty, \tag{19}$
thus,
$\langle v, w \rangle = 0. \tag{20}$
Note: It may be worth pointing out that the condition on $C$ 
$C^\dagger = (\bar C)^T = C; \tag{21}$
has implitions for $A$ and $B$ as in (8), for then
$\bar C = A - iB, \tag{22}$
$(\bar C)^T = A^T - iB^T; \tag{23}$
thus,
$A = A^T, \tag {24}$
$B = -B^T; \tag{25}$
we see that $A$, the real part of $C$, must be symmetric, whilst $B$, the imaginary part, must be skew-symmetric.  Finally we observe that the matrix $M$ (13) is in fact symmetric as well:
$M^T = \begin{bmatrix} A & -B \\ B & A \end{bmatrix}^T = \begin{bmatrix} A^T & B^T \\ -B^T & A^T \end{bmatrix} = \begin{bmatrix} A & -B \\ B & A \end{bmatrix} = M, \tag{26}$
by virtue of (24)-(25).  End of Note.
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