If $W, Y \in R^{n \times n}$, then how the eigenvectors and eigenvalues of these two matrices are related? $C = W +iY, B = \begin{bmatrix} W & -Y\\ Y & W\\ \end{bmatrix} $
Specifically, can we show that if $\lambda \in \lambda(C)$ is real, then $\lambda \in \lambda(B)$ ?
Let $u+iv$ be an eigenvector of $C$, with $u$ and $v$ real, and with eigenvalue $\lambda$. Then $$(W+iY)(u+iv)=\lambda(u+iv)$$ but also $$(W+iY)(u+iv)=Wu-Yv+i(Yu+Wv)$$ so $$Wu-Yv=\lambda u,\quad Yu+Wv=\lambda v$$ (This is where we use the hypothesis that $W,Y,u,v,\lambda$ are all real.) Then we get $$\pmatrix{W&-Y\cr Y&W\cr}\pmatrix{u\cr v\cr}=\pmatrix{Wu-Yv\cr Yu+Wv\cr}=\pmatrix{\lambda u\cr\lambda v}=\lambda\pmatrix{u\cr v\cr}$$