Are there any properties for matrix A (A= USVt) in Singular Value Decomposition?

Question

I have seen a lot of explanations about SVD with examples figure how it works. like for example figure in this link. by looking the figure it seems that m must be bigger than n (m>n) for matrix A with mxn. what about in real cases do we have to have a matrix which the rows must be wider than the columns?

Answer 1

SVD works for every matrix $A\in \mathbb{R}^{n \times m}$ and arbitrary $n, m$.

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As a comment to the case $m\leq n$: Consider this case and let be given a SVD for $A^T$, i.e.

$$ A^T = U S V^T. $$

Then $$A = V S^T U^T $$ is a SVD for $A$.