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Gaussian elimination reduces every finite matrix over a field to row echelon form
Statement
Every finite matrix over a field is row equivalent to a matrix in row echelon form. The reduction can be obtained by the Gaussian elimination algorithm.
Facts & Assumptions
Given: A matrix over a field.
Row swaps, nonzero row scalings and row additions are elementary row operations (Elementary row operations and row equivalence for finite matrices over a field).
Row echelon form requires successive leading entries to move right, with zeros below every leading entry and zero rows at the bottom (Row echelon form, reduced row echelon form, leading entries, pivots, pivot positions and pivot columns).
Induction applies to finite natural dimensions (The principle of mathematical induction).
Proof
If is not zero, locate the leftmost column containing a nonzero entry, swap one such entry into the first available row, scale it to , and use row additions to clear every entry below it. These are elementary operations and create the first pivot with all earlier columns zero.
Apply the same construction inductively to the strictly smaller submatrix below and to the right of that pivot. Operations confined to its rows do not change the earlier zero columns or pivot row, so the successive pivots move strictly right.
The procedure terminates when no row, no column or no nonzero entry remains; the unused rows are zero and lie at the bottom. The resulting matrix satisfies all echelon conditions, including the all-zero and zero-sized cases.
Depends on
Used by
- In positive dimension the determinant of an operator is computed from a representing matrix by row reduction, tracking swaps and row scalings Corollary
- A linear endomorphism of ℝⁿ sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant Theorem
- Gauss–Jordan elimination reduces every finite matrix over a field to reduced row echelon form Theorem
Dependency tree · two levels
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Sources
- D. Margalit and J. Rabinoff, Interactive Linear Algebra, §1.2 (standard reference, not scraped)
- J. Hefferon, Linear Algebra, 4th ed., Ch. One, §III.1 (standard reference, not scraped)