Idempotent matrices on quaternion algebra

Abstract

The aim of this study is to provide a complete and constructive classification of all 2×2 idempotent matrices defined over the quaternion algebra H. Using the scalar–vector decomposition of quaternionic elements, the idempotency condition M^2=M is transformed into a coupled system of scalar and vector equations that reveal the algebraic and geometric interrelations among the matrix entries. The classification covers three main structural families: (i) cases where the product of off-diagonal entries is real, (ii) degenerate triangular matrices with one vanishing off-diagonal term, and (iii) general configurations involving vector cross products and scalar constraints. The theoretical results are presented in parametric forms and illustrated with examples. These findings extend the classical theory of idempotent matrices to the non-commutative quaternionic framework, offering a geometric insight relevant to operator theory and 3D rigid-body kinematics.

Keywords

• non-commutative matrix theory
• quaternion algebra
• idempotent matrices

Ethical Statement

The authors declare that this study complies with established research and publication ethics.

Thanks

The authors are grateful to the reviewers and the editors for their valuable suggestions, which led to significant improvements in the manuscript.

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