WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 25, 2026
Coordinate-Independent Formulation of Matrix Division
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Abstract: This paper presents a coordinate-independent formulation of matrix division using the Levi-Civita tensor. It demonstrates that the co-divisor matrix concept, introduced for solving $$ AX = B $$ possesses a tensorial structure expressible via the antisymmetric symbol $$ \varepsilon_{i_1 \ldots i_n}. $$ The study provides a proof for the equality between the co-divisor matrix and the classical inverse product, i.e., $$ \frac{B}{A} = A^{-1}B. $$ A similar tensor representation is developed for row co-divisors, solving $$ XA = B. $$ This approach offers a geometrically meaningful framework, enabling a matrix-level generalization of Cramer’s rule and opening new perspectives for solving linear matrix equations.
Keywords:
Matrix division, Co-divisor matrix, Levi-Civita tensor, Coordinate-independent formulation, Cramer’s rule generalization, Linear matrix equations, Tensor representation, Antisymmetric tensor
Pages: 86-101
DOI: 10.37394/23206.2026.25.10