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?