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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.19308 |
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| _version_ | 1866913070073249792 |
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| author | Bhadala, Neha Behera, Ratikanta |
| author_facet | Bhadala, Neha Behera, Ratikanta |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19308 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure Preserving Algorithms for Quaternion Outer Inverses with Applications Bhadala, Neha Behera, Ratikanta Rings and Algebras 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. |
| title | Structure Preserving Algorithms for Quaternion Outer Inverses with Applications |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2506.19308 |