Rank function equations

Abstract

The purpose of this paper is to introduce the notion of rank function equation, and to present some results on such equations. In particular, we find all sequences $(A_{1}, ..., A_{k}, B)$ of nonzero nilpotent $n \times n$ matrices satisfying condition $$ \forall\, m \in \{1, ..., n\} :\, \sum_{i=1}^{k} r_{A_{i}}(m) = r_{B}(m),$$ and give a characterization of all sequences $(A_{1}, ..., A_{k}, B)$ of nilpotent $n \times n$ matrices such that $$ \forall\, m \in \{1, ..., n\} :\, \sum_{i = 1}^k f (r_{A_{i}} (m)) = r_{B} (m),$$ where $f : \mathbb{R} \supset [0, \infty) \longrightarrow \mathbb{R}$ is a function with certain natural properties. We also provide a geometric characterization of some solutions to rank function equations.