I am trying to find an $n \times m$ fat (i.e., $m > n$) matrix $T$ that solves
$$T^T T = X$$
where $X$ is a given $m \times m$ symmetric, positive semidefinite matrix.
I saw this post, but unfortunately it seems that no solution was yet found and $T$ in that case is squared, which is not my case.
If $T$ were square, one could use the Cholesky decomposition and find $Z$ such that $X = Z^T Z$. Unfortunately, I cannot do this since the Cholesky decomposition always produces a square $Z$.