Two direct solvers for a system of linear equations

TL;DR

This paper introduces two novel direct methods for solving linear systems that preserve the vector space structure and enable online computation, providing generalized inverses and null space projections.

Contribution

The paper presents two new direct solvers that maintain the vector space perspective and support online solutions, unlike traditional methods like QR or LU.

Findings

Methods produce generalized inverses and null space projections.

Both solutions support online, incremental data processing.

Approach respects the underlying vector space structure.

Abstract

A system of linear equations is normally understood as a linear mapping between two vector spaces. However, most direct solutions (e.g., QR, LU, ...) rely on the inelegant approach of back-substitution: a significant departure from such a characterization. In this paper, two new methods are introduced which respect the underlying vector space throughout, whether it be the row or column space of the coefficient matrix. Both solutions produce a generalized inverse as well as a projection operator for the null space. They also have the unique feature of admitting an online solution, which allows a solution to be computed as the data becomes available.