Proximal operator is defined for matrices as a map prox$_f:R^m\times R^n \rightarrow R^m\times R^n$:

prox$_f$(X) := argmin$_{Y\in R^m\times R^n}$ $ f(Y) + \frac{1}{2}||Y-X||^2$

In case of vectors, it is known http://arxiv.org/pdf/0912.3522v4.pdf that if $f(x) = \phi(x/p)$, $p \in R, p\ne0$, then prox$_f(x) = p$prox$_{\phi/p^2}(x/p)$.

Is there an alternative when we work with arbitrary matrices, i.e. not necessarily invertible? So for example I have $||PX||_1$, where $P \in R^m\times R^m $, $X \in R^m\times R^n$, and $X$ is a variable. It is known that the proximal mapping for $||X||_1$ is soft-thresholding operator, so I need to 'get rid of' $P$ inside the function.