Is $U = V$ in the SVD of symmetric matrices? Consider the SVD of matrix $A$:
$$A = U \Sigma V^\top$$
If $A$ is a symmetric, real matrix, is there a guarantee that $U = V$?
There is a similar question here that also posits $A$ is positive semi-definite. But I'm wondering whether $U$ would be equal to $V$ if $A$ is symmetric?

If $A=USU^T$ is a SVD, $A$ has to be positive semidefinite, as shown in another answer here.
The converse is not true. If $A$ is positive semidefinite and $A=USV^T$ is a SVD, $U$ is not necessarily equal to $V$. E.g. $A=U0V^T$ is a SVD for any two unitary matrices $U$ and $V$.
However, if $A$ is positive definite, we must have $U=V$ in its SVD, as shown in the answers to Is $U=V$ in the SVD of a symmetric positive semidefinite matrix?
