Rank Two Preservers on Symmetric Matrices with Zero Trace of Order at Most Four Over Real Numbers

Yeung, Kien Xin (2025) Rank Two Preservers on Symmetric Matrices with Zero Trace of Order at Most Four Over Real Numbers. Final Year Project (Bachelor), Tunku Abdul Rahman University of Management and Technology.

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Abstract

In this project, linear transformations that preserve matrix rank specifically rank two are investigated within the space of real symmetric matrices with zero trace of order at most four. The study is motivated by (Kok, 2021). The primary objective is to characterize all transformations that preserve the rank two property on this specific matrix space. This work derives necessary and sufficient conditions for a map to qualify as a rank two preserver. In particular, the study reveals that such maps are closely related to change of basis and congruence mappings. Examples and counterexamples are provided to illustrate key results. The findings contribute to a deeper understanding of the algebraic structures underlying low rank matrix transformations and lay the groundwork for future exploration in higher dimensional or complex matrix spaces.