Suppose I have a matrix $A \in \mathbb{R}^{m \times n}$ with singular value decomposition (SVD) $A = U S V^T$. I then stack $A$ on top of itself $k$ times:
$A' = [A; A; ...; A] \in \mathbb{R}^{km \times n}$
Is there any way to relate the SVD of $A'$ to the SVD of $A$? I'm equally happy with any result that goes either way (either $A$ to $A'$ or $A'$ to $A$).