Eigenvalue correspondence between Symplectic orthogonal & Unitary matrices

Question

We know $Sp(2n)\bigcap O(2n) \simeq U(n)$ via $\left(\matrix{A & B \\-B & A}\right)\mapsto A+iB$. Now if the eigenvalues of $A+iB$ are given as $e^{i\lambda_1},e^{i\lambda_2},\dots,e^{i\lambda_n}$, can we write down the corresponding eigenvalues of $\left(\matrix{A & B \\-B & A}\right)$ in terms of $\lambda_1,\lambda_2,\dots,\lambda_n?$

Answer 1

The identification that you are describing simply is an explicit identification of $\mathbb R^{2n}$ with $\mathbb C^n$ (basically $(x_1,\dots,x_n,y_1,\dots,y_n)\mapsto (x_1+iy_1,\dots,x_n+iy_n)$. Then $A+iB$ and $\begin{pmatrix} A & B \\ -B & A\end{pmatrix}$ describe the same linear map in the two pictures, and thus have the same eigenvalues over $\mathbb C$. Over $\mathbb R$, the only possible eigenvalues of an orthogonal matrix are $\pm 1$, and any occurrence of $0$ and $\pi$ among $\lambda_1,\dots,\lambda_n$ gives rise to a double real eigenvalue equal to $1$ respectively $-1$.