I am wondering if there exist transformations that preserve the rank of a matrix.
Specifically, if $A$ is a rank $k$ positive semi-definite matrix (PSD), can we exhibit a transformation $T$ such that $T(A)$ has rank $k$, and possibly is also PSD.
Moreover, is there a non-linear (convex?) function $f$ such that $B : B_{ij}=f(A_{ij})$ is PSD and of rank $k$ ?
$T(A)=CA$ or $T(A)=AC$ for any fixed square invertible matrix $C$, with dimensions compatible with the dimensions of $A$.
In particular when $C=cI=diag(c,...,c)$, $T(A)$ is $cA$ ($c$ assumed nonzero).
For your second question, there is a partial answer for full rank PSD matrices, i.e., PD matrices: take $f(x)=x^2$ because the Hadamard product of two PD matrices (here we take twice the same matrix) is still a PD matrix by Schur product theorem (https://en.wikipedia.org/wiki/Schur_product_theorem).