Spectrum of rotation matrices

Question

Given an $n$-tuple $(e^{i\theta_j})_{j=1}^n$ of $n$ points on the unit circle in $\mathbb{C}$, what are the necessary and sufficient conditions that guarantee the existence of a rotation matrix $R\in SO(n)$ such that the spectrum of $R$ is $\{e^{i\theta_1},\ldots,e^{i\theta_n}\}$?

I realize that $\sum_{j=1}^n\theta_j=0$ because of the constraint $\det R=1$. Is it also sufficient? Here is where I get stuck. In particular, if I form the spectral decomposition $Q=U\Lambda U^*$, where $\Lambda=diag(e^{i\theta_j})_{j=1}^n$ and U is some unitary matrix, how can I know that there always exists a $U$ such that $Q$ is real?

Answer 1

The complex (non-real) eigenvalues of a real matrix must come in conjugate pairs. Hence, a matrix $R \in \operatorname{SO}(n)$ must have a spectrum of the form

$$ z_1, \overline{z_1}, \dots, z_k, \overline{z_k}, x_1, \dots, x_{n-2k} $$

where $0 \leq 2k \leq n$, $|z_i| \in S^1 \setminus \{ \pm 1 \}$, $x_i \in \{ \pm 1 \}$ and $\prod_{i=1}^{n-2k} x_i = 1$. Conversely, given a list satisfying the conditions above, one can always build a matrix $R$ is precisely that list by taking the block diagonal matrix

$$ R = \operatorname{diag}(R(z_1), \dots, R(z_k), x_1, \dots, x_{n-2k}) $$

where each $R(z_i) \in \operatorname{SO}(2)$ is the $2 \times 2$ rotation matrix given by

$$ R(z_i) = \begin{pmatrix} \Re(z_i) & -\Im(z_i) \\ \Im(z_i) & \Re(z_i) \end{pmatrix}. $$

Answer 2

A diagonalizable complex matrix is similar to some matrix with real entries if and only if its eigenvalues come in conjugate pairs.

In your notation, you must be able to partition your $\theta$'s into pairs of the form $\pm \theta$, except in the case where $\theta = 0$.

If we were looking for elements of $O(n)$, we could also have an odd number of $\theta$s equal to $\pi$.