Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that:

A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric matrix, there is a unitary matrix $W \in U(n)$ such that $W A W^T$ is a real non-negative diagonal matrix.

I wonder whether there are similar statements for $$W \in \frac{SU(n) \times \mathbb{Z}_{n k}}{\mathbb{Z}_n} \subset U(n),$$ where the mod factor ${\mathbb{Z}_n}$ is due to that:

(1) This ${\mathbb{Z}_n}$ is both the center of $SU(n)$.

(2) This ${\mathbb{Z}_n}$ is also the normal subgroup of $\mathbb{Z}_{n k}$, which is along the diagonal $U(1)$.

My questions, in other words, are that

If $A$ is a rank-$n$ complex symmetric matrix, for any unitary matrix $W \in \frac{SU(n) \times \mathbb{Z}_{n k}}{\mathbb{Z}_n} $, can we always make that $W A W^T$ always a diagonal matrix? What will be the form of
$$W A W^T,$$ if it is diagonal? How to find such a $W$?

I suppose there is an additional complex phase $e^{i \theta}$ in addition to a real non-negative diagonal matrix called $D$ $$W A W^T =e^{i \theta} D,$$ are there some restrictions about $e^{i \theta}$? (e.g. when $W \in U(n)$, we can always set it to $e^{i \theta}=1$. )

I am particularly interested in when

$n=2$ and $n=3$. So we can simplify to these two particlar cases.


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