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][1], 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 $Z$.


  [1]: https://mathoverflow.net/questions/78106/solving-a-quadratic-matrix-equation