Let $A$ be a real $m\times n$ matrix. Suppose $m_1 + m_2 = m$, and construct $A_1$ by taking the 1st to $m_1$th rows of $A$. Similarly, one can construct $A_2$ by taking the $(m_1 + 1)$th to $m$th row of $A$.
In order words, $A = \begin{bmatrix} A_1\\ A_2 \end{bmatrix}$, where $A_1$ is $m_1 \times n$, and $A_2$ is $m_2 \times n$.
By SVD, $A = UDV^T$, where $U$ and $V$ are orthogonal matrices, and $D$ is a diagonal matrix. Similarly, we can also have $A_1 = U_1D_1V^T_1$ and $A_2 = U_2D_2V_2^T$ by SVD.
Clearly, I don't think $U = \begin{bmatrix} U_1\\ U_2 \end{bmatrix}$ or $V = \begin{bmatrix} V_1\\ V_2 \end{bmatrix}$. My question is, how does $U$ relate to $U_1, U_2$? $D$ to $D_1, D_2$? And $V$ to $V_1, V_2$?