Primal Topologies on Finite-Dimensional Vector Spaces Induced by Matrices

Abstract

Given an matrix $A$ , considered as a linear map $A:{ℝ}^n⟶{ℝ}^n$ , then $A$ induces a topological space structure on ${ℝ}^n$ which differs quite a lot from the usual one (induced by the Euclidean metric). This new topological structure on ${ℝ}^n$ has very interesting properties with a nice special geometric flavor, and it is a particular case of the so called “primal space,” In particular, some algebraic information can be shown in a topological fashion and the other way around. If $X$ is a non-empty set and $f:X⟶X$ is a map, there exists a topology ${\tau }_f$ induced on $X$ by $f$ , defined by ${\tau }_f=\left\{U\subset X:f^{-1}\left(U\right)\subset U\right\}$ . The pair $\left(X,{\tau }_f\right)$ is called the primal space induced by $f$ . In this paper, we investigate some characteristics of primal space structure induced on the vector space ${ℝ}^n$ by matrices; in particular, we describe geometrical properties of the respective spaces for the case.