Jacobians of singular matrix transformations: Extensions
Jose A. Diaz-Garcia, Ram\'on Gutierrez-Sanchez
TL;DR
This paper develops a unified framework for computing Jacobians of various singular matrix transformations across different algebraic systems, including real, complex, quaternion, and octonion cases, with respect to Hausdorff measure.
Contribution
It introduces a comprehensive method to derive Jacobians for singular matrix transformations across multiple algebraic structures, extending previous isolated results.
Findings
Unified Jacobian formulas for singular matrix transformations
Applicable to real, complex, quaternion, and octonion matrices
Provides a systematic approach for Hausdorff measure-based calculations
Abstract
This article presents a unified approach to simultaneously compute the Jacobians of several singular matrix transformations in the real, complex, quaternion and octonion cases. Formally, these Jacobians are obtained for real normed division algebras with respect to the Hausdorff measure.
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Taxonomy
TopicsMatrix Theory and Algorithms · Advanced Topics in Algebra · Algebraic and Geometric Analysis