Numerical solution of a class of quasi-linear matrix equations

Given the matrix equation $\mathbf{A}\mathbf{X}+\mathbf{X}\mathbf{B}+f(\mathbf{X})\mathbf{C}=\mathbf{D}$ in the unknown $n\times m$ matrix $\mathbf{X}$, we analyze existence and uniqueness conditions, together with computational solution strategies for $f:{\mathbb{R}}^{n\times m}\to \mathbb{R}$ being a linear or nonlinear function. We characterize different properties of the matrix equation and of its solution, depending on the considered classes of functions $f$. Our analysis mainly concerns small dimensional problems, though several considerations also apply to large scale matrix equations.