| dc.description.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. | en |
