Determinantal representations of the Moore-Penrose inverse over the quaternion skew field and corresponding Cramer's rules

TL;DR

This paper develops determinantal formulas for the Moore-Penrose inverse over quaternions, enabling Cramer's rule-based solutions for quaternionic linear systems, expanding classical linear algebra into quaternionic contexts.

Contribution

It introduces a novel determinantal representation of the Moore-Penrose inverse over quaternions and derives corresponding Cramer's rules for least squares solutions.

Findings

Determinantal formulas for quaternionic Moore-Penrose inverse derived

Cramer's rules established for quaternionic linear systems

Extension of classical linear algebra concepts to quaternion skew fields

Abstract

Determinantal representation of the Moore-Penrose inverse over the quaternion skew field is obtained within the framework of a theory of the column and row determinants. Using the obtained analogs of the adjoint matrix, we get the Cramer rules for the least squares solution of left and right systems of quaternionic linear equations.