Rank function equations. (English) Zbl 1300.15007
Given a square matrix \(A\) of order \(n\), consider the notation \(r_A(m):=\mathrm{rank}\left(A^m\right)\), where \(m\) is a positive integer. This paper introduces the definition of rank function equations as being a system of equations of the form
\[
f\left(r_{A_1}(m)\right)+\cdots+f\left(r_{A_k}(m)\right)=g\left(r_{B}(m)\right),\;\forall m\in S,
\]
where the unknowns are the square matrices \(A_1,\dots,A_k, B\) of order \(n\), \(f\) and \(g\) are functions over the positive integers and \(S\) is a set of positive integers. The nilpotent solutions of some particular rank function equations are found or characterized.
Reviewer: João R. Cardoso (Coimbra)
MSC:
15A24 | Matrix equations and identities |
14M12 | Determinantal varieties |
15A03 | Vector spaces, linear dependence, rank, lineability |