I am learning how to compute SVD now and I ran across a question. Why does it actually matter whether $m \leq n$ or $m>n$ in computing the SVD?
As I read through the motivation of SVD I found out that if $A = U\Sigma V^T$ the eigenvectors of $AA^T$ corresponds to the orthogonal matrix $U$, while the eigenvectors of $A^TA$ corresponds to the orthogonal matrix $V$. And since $Av_i = \sigma_i u_i$ and $A^Tu_i = \sigma_iv_i$ I do not see any reason we must enforce the 2 cases. Am I missing anything here?