The Infinite Gauss-Jordan Elimination on Row-Finite \omega\ x \omega\ Matrices

TL;DR

This paper extends the Gauss-Jordan elimination algorithm to infinite row-finite matrices, enabling constructive solutions for infinite linear systems while preserving row equivalence, offering an alternative to classical existence results.

Contribution

It introduces a new infinite Gauss-Jordan elimination method based on rightmost pivots, providing a constructive approach to Quasi-Hermite forms without relying on the axiom of countable choice.

Findings

The algorithm reduces infinite matrices to a lower row-reduced form.

It ensures the general solution of infinite linear systems is fully constructible.

Provides an alternative to classical existence and uniqueness proofs.

Abstract

The Gauss-Jordan elimination algorithm is extended to reduce a row-finite $\omega\times\omega$ matrix to lower row-reduced form, founded on a strategy of rightmost pivot elements. Such reduced matrix form preserves row equivalence, unlike the dominant (upper) row-reduced form. This algorithm provides a constructive alternative to an earlier existence and uniqueness result for Quasi-Hermite forms based on the axiom of countable choice. As a consequence, the general solution of an infinite system of linear equations with a row-finite coefficient $\omega\times\omega$ matrix is fully constructible.