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. I would like to know the following:

1. The operator norm as a function of the norm of $ U $.

2. The eigenvalues and eigenspaces (specifically invariant subspaces).

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

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. I realize operator theory and some advanced functional analysis might be my salvation, but unfortunately, I lack knowledge in these two areas. All help is kindly appreciated.

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.