Primal Topologies on Finite-Dimensional Vector Spaces Induced by Matrices

Abstract

Given an matrix A, considered as a linear map A:ℝn⟶ℝn, then A induces a topological space structure on ℝn which differs quite a lot from the usual one (induced by the Euclidean metric). This new topological structure on ℝn 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 τf induced on X by f, defined by τf=U⊂X:f−1U⊂U. The pair X,τf is called the primal space induced by f. In this paper, we investigate some characteristics of primal space structure induced on the vector space ℝn by matrices; in particular, we describe geometrical properties of the respective spaces for the case.