%0 Journal Article
%T Rank function equations
%A Piotr Pokora
%A Marcin Skrzy¨˝ski
%J Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
%D 2012
%I Krak¨®w : Wydawnictwo Naukowe Uniwersytetu Pedagogicznego
%X 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.
%U http://studmath.up.krakow.pl/index.php/studmath/article/view/116