Projective lines over Jordan systems and geometry of Hermitian matrices

TL;DR

This paper explores the relationship between projective lines in ring geometry and projective spaces in matrix geometry derived from sets of Hermitian matrices over fields with involution, showing they are fundamentally connected.

Contribution

It demonstrates that projective lines and projective spaces constructed from Hermitian matrices over fields with involution are essentially the same set of points, clarifying their geometric relationship.

Findings

Projective lines and spaces from Hermitian matrices are equivalent.

The two geometric concepts are based on the same set of points.

The paper clarifies the notational differences between the concepts.

Abstract

Any set of $\sigma$-Hermitian matrices of size $n \times n$ over a field with involution $\sigma$ gives rise to a projective line in the sense of ring geometry and a projective space in the sense of matrix geometry. It is shown that the two concepts are based upon the same set of points, up to some notational differences.