I am trying to find properties or constraints on a $(p \times n) $matrix $U$ such that upon left multiplying a real symmetric square positive semi-definite matrix $V$ with $U$ the resulting matrix $W$ of dimensions $(p \times n) $ is still real positive semi-definite matrix. In other words i am trying to establish that there is $U$ such that $$W = U V$$ where $W$ and $V$ are real positive semi-definite matrices.
If $W=V$ then obviously $U=I$ fits the equation.
I have a suspicion that this should be possible as by definition of real symmetric positive semi-definite matrix, $V$ should have non-negative eigenvalues, which implies that it should have a right inverse which I suspect would also be positive semi-definite matrix, so if $W$ (some positive semi-definite matrix) is multiplied with the right inverse we should be able to compute $U$.
Any help would be much appreciated.
edit I suspect if my reasoning about computing $U$ above is correct then it should also be positive semi-definite matrix. So perhaps a way to construct $U$ may be by applying Gram-Schmidt process and finding new eigenvectors for W.