[Paper Review] An Explicit Construction of Gauss-Jordan Elimination Matrix

Yi Li|ArXiv.org|Jul 29, 2009

Spectroscopy and Chemometric Analyses6 references3 citations

TL;DR

This paper presents an explicit determinant-based construction for entries in the Gauss-Jordan elimination process, showing that each off-diagonal entry $ a^k_{ij} $ (with $ i \neq j, j > k $) in the matrix after $ k $ steps can be expressed as a ratio of two determinants derived from the original matrix $ A $. The method provides a new generalized Cramer's rule applicable to arbitrary systems, including those with parametric coefficients, and enables direct solution construction for $ AX = b $.

ABSTRACT

A constructive approach to get the reduced row echelon form of a given matrix $A$ is presented. It has been shown that after the $k$th step of the Gauss-Jordan procedure, each entry $a^k_{ij}(i<>j; j > k)$ in the new matrix $A^k$ can always be expressed as a ratio of two determinants whose entries are from the original matrix $A$. The new method also gives a more general generalization of Cramer's rule than existing methods.

Motivation & Objective

• To develop a constructive method for computing the reduced row echelon form of a matrix using determinants.
• To generalize Cramer’s rule beyond existing formulations, particularly for systems with parametric coefficients.
• To provide a direct computational pathway for solving $ AX = b $ using determinant expressions.
• To establish a theoretical framework where Gauss-Jordan elimination steps can be explicitly represented via determinants from the original matrix.

Proposed method

• The method expresses each entry $ a^k_{ij} $ in the matrix after $ k $ Gauss-Jordan steps as a ratio of two determinants derived from the original matrix $ A $.
• It uses recursive determinant identities based on Bareiss’s fraction-free Gaussian elimination to derive expressions for entries at each elimination stage.
• The approach distinguishes three cases based on the row index $ i $: $ i > k $, $ i = k $, and $ i < k $, each with a distinct determinant formula.
• For $ i < k $, the method incorporates a sign factor $ (-1)^{k-i+1} $ in the expression to account for row permutations and determinant expansion.
• The derivation relies on the identity $ \left| \begin{smallmatrix} M & U \\ R & a \end{smallmatrix} \right| = \text{det}(M) \cdot \text{det}(\text{submatrix}) $, enabling recursive determinant decomposition.
• The method ensures all entries in $ A^k $ are expressed as rational functions of original matrix entries, with denominators being principal minors.

Experimental results

Research questions

• RQ1 Can the entries of a matrix during Gauss-Jordan elimination be explicitly expressed using determinants from the original matrix?
• RQ2 How can Cramer’s rule be generalized to apply to systems with parametric coefficients or non-square matrices?
• RQ3 Is there a uniform determinant-based formula for all off-diagonal entries $ a^k_{ij} $ with $ j > k $, regardless of row position?
• RQ4 Can this determinant-based approach directly yield a solution to $ AX = b $ without back-substitution?

Key findings

• After $ k $ Gauss-Jordan elimination steps, each entry $ a^k_{ij} $ with $ i \neq j $ and $ j > k $ can be written as a ratio of two determinants from the original matrix $ A $.
• The expression for $ a^k_{ij} $ depends on the relative row index $ i $: for $ i > k $, it is $ \frac{a^{(k)}_{ij}}{a^{(k-1)}_{kk}} $; for $ i = k $, it is $ \frac{a^{(k-1)}_{ij}}{a^{(k-1)}_{kk}} $; and for $ i < k $, it includes a sign factor $ (-1)^{k-i+1} $.
• The method generalizes Cramer’s rule in a way that is distinct from prior approaches, enabling direct solution construction for $ AX = b $.
• The approach is valid over any commutative ring, provided leading principal minors are non-zero, extending its applicability beyond real or complex matrices.
• The recursive structure of the determinant expressions mirrors Bareiss’s fraction-free elimination, ensuring exact arithmetic without floating-point errors.
• The method supports applications in parametric linear systems and integer programming by enabling polynomial interpolation on determinant-based solutions.