| Summary: | 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 $n imes n$ 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 $n imes n$ 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.
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