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Gauss–Jordan elimination reduces every finite matrix over a field to reduced row echelon form
Statement
Every finite matrix over a field is row equivalent to a matrix in reduced row echelon form. The reduction can be obtained by Gauss–Jordan elimination.
Facts & Assumptions
Given: A finite matrix over a field.
Gaussian elimination row reduces every finite matrix to row echelon form (Gaussian elimination reduces every finite matrix over a field to row echelon form).
Reduced row echelon form requires every pivot to be and the only nonzero entry in its column (Row echelon form, reduced row echelon form, leading entries, pivots, pivot positions and pivot columns).
Proof
Use [L1] to reach row echelon form, and scale every nonzero row by the inverse of its pivot so that each pivot is .
Process the pivots from bottom to top. For a pivot in column , add suitable multiples of its row to all rows above it to clear column ; later pivot columns remain unchanged because the current row is zero in those columns.
Echelon order and the zero rows are preserved, every pivot is , and every pivot column is zero off its pivot. Thus the output is reduced row echelon form, including when there is no pivot.
Depends on
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 12 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
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)