The singular value decomposition of a complex matrix $A$ can be written as
$$ A= UDV^{H} $$
where $U$ and $V$ are hermitian matrices and $D$ is a diagonal matrix with entries $(D)_{ii}=\sigma_i\geq0$ called the singular values. My question is whether we can make a statement about the matrices $U$ and $V$ and the singular values, if the matrix $A$ is transformed according to
$$ (A)_{ij} \rightarrow (A)_{ij} e^{\iota \alpha_{ij}} $$
where $\iota$ is the imaginary unit. Furthermore, the phase matrix $\alpha$ should be symmetric and real $(\alpha)_{ij}=(\alpha)_{ji}\in\mathbb{R}$.
From a physicist's point of view, I would expect that the singular values remain unchanged under such a transformation because the singular value decomposition can be used to rewrite entangled states/wave functions. Furthermore, the moduli squared of the columns of $U$ and rows of $V^H$ should also be unchanged, or rather, it should be possible to find a decomposition that guarantees this.
These results seem to be quite elementary from a physics point of view so maybe there are already references that analyze this.