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Primal Topologies on Finite-Dimensional Vector Spaces Induced by Matrices

Abstract:

Given an matrix A, considered as a linear map A:ℝnℝn, 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:XX is a map, there exists a topology τf induced on X by f, defined by f=UX:f-1UU. The pair Xf 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.