The (orbifold) space of symmetric complex matrix 2

Question

Let $K$ be a rank-2 symmetric complex matrix, such that the transpose $K^T=K$ is itself.

Modifying the previous attempt, let $V$ be a rank-2 matrix in the subset of unitary matrix, $$V \in \frac{SU(2) \times \mathbb{Z}_n}{\mathbb{Z}_2} \subset U(2)$$ where $n$ is an even integer and here we mod out the common normal subgroup ${\mathbb{Z}_2}$ out of numerator group. Here the the $ \mathbb{Z}_n \subset U(1) \subset U(2)$ is the diagonal subgroup generated by the $n$-root of $1$.

Consider the identification between any K and K' of any rank-2 symmetric complex matrix, $$ K\sim K', $$ if it satisfies $$ V^T K V =K', $$ for any $V \in \frac{SU(2) \times \mathbb{Z}_n}{\mathbb{Z}_2}.$

question:

1. What is the real dimension of the new space of $K$ (under the $K\sim K'$ and $V^T K V =K'$, for any $V \in \frac{SU(2) \times \mathbb{Z}_n}{\mathbb{Z}_2}$ condition)?

2. How do we parametrize this new space of $K$ in terms of a rank-2 matrix (mod out the redundancy under the $K\sim K'$ and $V^T K V =K'$, for any $V \in \frac{SU(2) \times \mathbb{Z}_n}{\mathbb{Z}_2}$ condition)?

(p.s. This space may be a called an orbifold space(?). i.e. The (orbifold) space of symmetric complex matrix after mod out a relation identifying a unitary matrix.)

3. We can also consider $V$ be a rank-3 matrix in the subset of unitary matrix, $$V \in \frac{SU(3) \times \mathbb{Z}_n}{\mathbb{Z}_3} \subset U(3)$$ where $n$ is a mod 3 = 0 integer and here we mod out the common normal subgroup ${\mathbb{Z}_3}$ out of numerator group. If so, what is the simplest form of the $K$ to parametrize $K$ after identification $K\sim K'$ and $V^T K V =K'$?

Answer 1

Autonne-Takagi factorization says that for any symmetric complex matrix $K$ there exists a unitary matrix $V \in U(N)$ with $V^T K V = D$, where $D$ is a diagonal matrix with the non-negative real values. (They are the square roots of the eigenvalues of $K K^*$ along the diagonal.)

If we apply Autonne-Takagi factorization, we can obtain the diagonal matrix with the non-negative real values, say $$ diag(v_1, \dots, v_n). $$

Once we limit ourselves to only $$V \in \frac{SU(N)\times \mathbb{Z}_n}{\mathbb{Z}_N},$$ we should only get an

Answer: $$ e^{i \theta}diag(v_1, \dots, v_n), $$ with $$ \theta\in [0, 2\pi/n), $$

where $\theta=2\pi/n$ can be identified with $\theta=0$, under a proper subset of diagonal $V \in \mathbb{Z}_n \subseteq \frac{SU(N)\times \mathbb{Z}_n}{\mathbb{Z}_N}.$