I have some real matrix $A \in \mathbb{R}^{m \times n}$ with SVD $A = U S V^T$. If I splinter (probably not the correct mathematical term) the matrix into its rank-1 rows $A_i \in \mathbb{R}^{1 \times n}$ and compute the SVD of each row separately $A_i = u_i s_i v_i^T$, how do the singular values and singular vectors of the original matrix relate to the singular values and singular vectors of each row's SVD?
Edit: If $U \in \mathbb{R}^{m \times m}, S \in \mathbb{R}^{m \times n}, V \in \mathbb{R}^{n \times n}$, I believe I can write the $i$th row vector of $A$ as
$$A_i = [\sum_{k=1}^{rank(A)} s_k U_{ik} V_{1k}, ..., \sum_{k=1}^K s_k U_{ik} V_{nk}]$$
Is that correct? If so, does that help?