Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity

Year 2021, Volume: 4 Issue: 4, 147 - 153, 30.12.2021

Maria İsabel Garcia-planas

https://doi.org/10.32323/ujma.983509

Abstract

In this paper pairs of matrices under similarity are considered because of their scientific applications, especially pairs of matrices being simultaneously diagonalizable. For example, a problem in quantum mechanics is the position and momentum operators, because they do not have a shared base representing the system's states. They do not commute, and that is why switching operators form a crucial element in quantum physics. A study of the set of linear operators consisting of pairs of simultaneously diagonalizable matrices is done using geometric constructions such as the principal bundles. The main goal of this work is to construct connections that allow us to establish a relationship between the local geometry around a point with the local geometry around another point. The connections give us a way to help distinguish bundle sections along tangent vectors.

Keywords

Bundles, simultaneous similarity, connection, curvature

Supporting Institution

No support

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