Uniqueness of $UV^T$ of non-singular matrix $A$

Question

Let $A$ be a $n \times n$ non-singular matrix. Then we can take many alternative SVD from from $A$. Like $A = U_1 \Sigma V_1^\top = U_2 \Sigma V_2^\top = \cdots$ where $\Sigma$ is an ordered diagonal (invertible) matrix of singular values. Can we say that $U_1V_1^\top = U_2V_2^\top = \cdots$ ? I.e., I am curious about the uniqueness of $UV^T$.