Solution of Linear Systems of Equations Ax=b and Ax=0 using Unifying Approach with Geometric Algebra: Outer Product Application and Angular Conditionality

TL;DR

This paper introduces a unified geometric algebra approach using the outer product to solve linear systems Ax=b and Ax=0, providing analytical solutions and a new method for estimating matrix conditionality, including non-square matrices.

Contribution

It presents a novel geometric algebra-based formulation for solving linear systems and estimating matrix conditionality, extending applicability to non-square matrices.

Findings

Provides an analytical solution for linear systems using outer product

Introduces a new approach to matrix conditionality estimation

Applicable to both square and non-square matrices

Abstract

A solution of linear systems of equations Ax=b and Ax=0 is a vital part of many computational packages. This paper presents a novel formulation based on the projective extension of the Euclidean space using the outer product (extended cross-product). This approach enables to solve the both cases, i.e. Ax=b and Ax=0. The proposed approach leads actually to an analytical solution of linear systems in the form on which the other vector operation can be applied before using the numerical evaluation. This contribution also proposes a new approach to the conditionality estimation of matrices applicable to non-squared matrices. It splits the conditionality to structural conditionality showing matrix property if nearly unlimited precision is used, numerical issue which depends on numerical representation with respect to the right-hand side influence, if given.