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It seems that two different algorithms are referred to as the "Gauss-Jordan elimination".

  • Which one is the one?
  • Is any of the two algorithms used in modern numerical linear algebra for RREF?
  • Am I right to think that one of the algorithms might have better numerical properties?

Gauss-Jordan variant 1. This variant starts as the standard Gaussian elimination: the multiples of rows of $\boldsymbol{A}\in\mathbb{R}^{m\times n}$ are subtracted from rows below to create zeros, each such operation is expressed as a lower uni-triangular $\boldsymbol{L}_{k}$, so we obtain $$\boldsymbol{L}_{m}\cdots\boldsymbol{L}_{1}\boldsymbol{A}=\boldsymbol{U},$$ where $\boldsymbol{U}$ is upper-triangular and also in the row echelon form (but not necessarily in RREF). For simplicity, I will omit row exchanges. So far we've got the LU-decomposition, $\boldsymbol{A}=\boldsymbol{LU}$ with $\boldsymbol{L}:=(\boldsymbol{L}_{m}\cdots\boldsymbol{L}_{1})^{-1}$. We can also make all pivots in $\boldsymbol{U}$ equal $1$, and write $\boldsymbol{A}=\boldsymbol{LDU}$ with some other $\boldsymbol{U}$ and $\boldsymbol{L}$.

Now we continue with the "back-substitution" step: we subtract multiples of rows in $\boldsymbol{U}$ from rows above in order to create zeros above every pivot. The creation of zeros above one pivot is expressed as an upper uni-triangular $\boldsymbol{U}_k$, so we have $$(\boldsymbol{U}_{r}\dots\boldsymbol{U}_{1}\boldsymbol{D}\boldsymbol{L}_{m}\cdots\boldsymbol{L}_{1})\boldsymbol{A}=\boldsymbol{E}_{\boldsymbol{A}},$$ where $\boldsymbol{E}_{\boldsymbol{A}}$ is the unique reduced row echelon form (RREF) of ${\boldsymbol{A}}$.

[On a side note: the inverse of the parenthesis on the left-hand side should give us the basic columns of ${\boldsymbol{A}}$, but we can take them directly from ${\boldsymbol{A}}$ by looking at the pivot's positions and recalling how we permuted the rows, right?]

Gauss-Jordan variant 2. In contrast to what we did in the standard Gaussian elimination, now, at each step we subtract a multiple of the current row from all other rows to create zeros both above and below the pivot. Each such operation is expressed as $\boldsymbol{T}_{k}$, which is neither upper- nor lower-triangular. After including scaling of rows to make pivotal elements equal to $1$, we obtain $$\boldsymbol{T}_{r}\cdots\boldsymbol{T}_{1}\boldsymbol{A}=\boldsymbol{E}_{\boldsymbol{A}}.$$

Different sources define different variants. For example,

  • variant 1 is referred to as the Gauss-Jordan method in

    1. Wikipedia's article on Gaussian elimination, and also in
    2. Gilbert Strang's Linear Algebra and Its Applications, 4th ed (2006), p. 53.

  • Variant 2 is said to be the Gauss-Jordan method in

    1. Carl Meyer's Matrix Analysis and Applied Linear Algebra (2000), p. 47.
paperskilltrees
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    The language is not standard. In Friedberg, Insel, and Spence for example, they refer to something like variant 1 (yielding RREF) as Gaussian elimination, and something like variant 2 as Gauss-Jordan elimination. They claim for large matrices the former requires ~50% fewer operations than the latter, but would still require modification to avoid roundoff errors. – blargoner Sep 18 '22 at 03:16
  • @blargoner It turns out there are several books on linear algebra and editions thereof by these authors. Could specify which one you were referring to? Thanks! (Also the page numbers, if available?) – paperskilltrees Sep 18 '22 at 14:44
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    I was referencing their Linear Algebra, 4th ed., 2003, pp. 186-7. – blargoner Sep 18 '22 at 15:47
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    I am tutoring a class on numerics and I was wondering the same. Because the professor introduces two algorithms called Gauss elimination and Gauss–Jordan elimination, while the latter is supposed to be worse and it was not really clear to me, why we should teach it then. And I was not able to find any reference explaining these differences. https://math.stackexchange.com/questions/4436533/algorithmical-difference-between-gauss-and-gauss-jordan-elimination – Daniel Apr 28 '25 at 10:14

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