In my studies of applied analysis and applied linear algebra, this interesting problem and concept came up:

Let us consider the space of all $ m \times n $ real matrices, and define a scalar function $ \phi : \mathbb{R} \to \mathbb{R} $ as $ \phi(x)=\max{(x,0)} $ and we extend this definition to matrices by applying $ \phi $ entrywise, meaning $ (\phi(A))_{i,j} = \phi(A_{i,j}) $. Now suppose we have a real matrix of fixed dimensions $ U \in \mathbb{R} ^{r \times m} $ which we view as a linear operator on the matrix space above, and we define the commutator operator of $ \phi $ and $ U $ for all $ X \in \mathbb{R} ^{m \times n} $ as: \begin{align*} [\phi, U](X) = \phi(UX)-U\phi(X) \end{align*}

I wish to characterize this commutator and study it in as much detail as possible, but I seem to be unsuccessful in approaching the questions which came up, as this operator is nonlinear and beyond my knowledge. I would like to know the following:

1. Matrices $ X $ for which $ [\phi,U](X)=0 $ and their characterization.

2. Equivalent representations of the commutator which might ease analysis (perhaps in series form with known convergence regions).

3. Dimension-dependent bounds on the norm of image matrices.

This problem arose in my research of theoretical data science (dimensionality reduction), and I certainly would appreciate help on the questions raised as I find myself without a way to proceed due to the nonlinearity. I realize operator theory and some advanced functional analysis relating to nonlinear operator theory might be my salvation, but unfortunately, I lack knowledge in these two areas. All help is kindly appreciated.