2025/06/24 by Neha Bhadala, Ratikanta Behera, Bhadala, Neha +1
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Algebraic and Geometric Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2506.19308
openalex publication_date 2025/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This study investigates the theoretical and computational aspects of quaternion generalized inverses, focusing on outer inverses and 1,2-inverses with prescribed range and/or null space constraints. In view of the non-commutative nature of quaternions, a detailed characterization of the left and right range and null spaces of quaternion matrices is presented. Explicit representations for these inverses are derived, including full rank decomposition-based formulations. We design two efficient algorithms: one leveraging the Quaternion Toolbox for MATLAB (QTFM), and the other employing a complex structure preserving approach based on the complex representation of quaternion matrices. With suitable choices of subspace constraints, these outer inverses unify and generalize several classical inverses, including the Moore-Penrose inverse, the group inverse, and the Drazin inverse. The proposed methods are validated through numerical examples and applied to two real-world tasks: quaternion-based color image deblurring, which preserves inter-channel correlations, and robust filtering of chaotic 3D signals, demonstrating their effectiveness in high-dimensional settings.