I had a question about SVD. Specifically about the size of matrices $U$, $\Sigma$ and $V$ decomposed from the $m\times n$ matrix $X$ using the formula
$$X = U \Sigma V^T $$
Most of the the tutorial literature says that the resulting sizes are
- $U$ is $m \times m$
- $\Sigma$ is $m \times n$
- $V$ is $n \times n$
However, there have been quite few times when the sizes given are
- $U$ is $m \times n$
- $\Sigma$ is $n \times n$
- $V$ is $n \times n$
In other words instead of $\Sigma$ being the matrix with possibly different number of rows and columns, its $U$ with the different number of rows and columns.
The math works out, so why (and in what cases) is this less frequent version used?