Determinantal representations of the quaternion weighted Moore-Penrose inverse and corresponding Cramer's rule

TL;DR

This paper develops determinantal formulas and Cramer's rule for the weighted Moore-Penrose inverse of quaternion matrices, extending classical linear algebra concepts to the noncommutative quaternion setting.

Contribution

It introduces new determinantal representations and Cramer's rules for quaternion weighted Moore-Penrose inverses using noncommutative determinants.

Findings

Derived weighted singular value decomposition for quaternion matrices.

Established limit and determinantal representations of the inverse.

Formulated Cramer's rule for quaternion linear systems.

Abstract

Weighted singular value decomposition (WSVD) and a representation of the weighted Moore-Penrose inverse of a quaternion matrix by WSVD have been derived. Using this representation, limit and determinantal representations of the weighted Moore-Penrose inverse of a quaternion matrix have been obtained within the framework of the theory of the noncommutative column-row determinants. By using the obtained analogs of the adjoint matrix, we get the Cramer rules for the weighted Moore-Penrose solutions of left and right systems of quaternion linear equations.