I was trying to solve the problem of finding the value of a non-squared matrix $T$ ($n \times m$) which solves

$$ T^T T = X$$

where $X$ is a symmetric and positive semidefinite $m \times m$ matrix, and $m > n$.

I saw this post (Solving a quadratic matrix equation) but unfortunately it seems that no solution was yet found and $T$ in that case is squared, which is not my case.

I know that if $T$ would be squared, one could use the Cholesky decomposition and find that $X = Z^T Z$ and assign $T = Z$. Unfortunately I cannot do this since the Cholesky decomposition always gives squared $Z$.