Rank function equations

Piotr Pokora, Marcin Skrzyński

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.

Keywords


rank function equation; rank function; conjugacy class; nilpotent matrix; Jordan partition

Mathematics Subject Classification


15A24; 14M12

References


Gantmacher, F.R. "Théorie des matrices." Paris: Dunod, 1966.

Gerstenhaber, M. "On dominance and varieties of commuting matrices." Ann.Math. 73 (1961): 324-348.

Shafarevich, I.R. "Basic algebraic geometry." Berlin-New York: Springer-Verlag, 1977.

Skrzyński, M. "Rank functions of matrices." Univ. Iagel. Acta Math. 37 (1999): 139-149.


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