What is the use of reduced row echelon form (not a row echelon form)?

Question

There is raw echelon form and reduced raw echelon form: https://en.wikipedia.org/wiki/Row_echelon_form https://en.wikipedia.org/wiki/Row_echelon_form#Reduced_row_echelon_form

The row echelon form is used to solved the system of linear equations. What is the practical use of reduced raw echelon form?

Previous question in the forum was related to row echelon form (and not to reduced row echelon form): Is reducing a matrix to row echelon form useful at all?

Thanks

@amWhy This question was marked as duplicate in error. – Aug 19 '17 at 13:28 — , Aug 19 '17 at 13:28
Can you use please remove the "duplicate" from the subject? – student Aug 19 '17 at 13:45 — student, Aug 19 '17 at 13:45

score 3 · Accepted Answer · answered Aug 19 '17 at 14:07

For the resolution of linear systems, Gaussian elimination is preferred over Gauss-Jordan (these parallel the echelon and reduced echelon forms) as the former involves less operations (roughly $n^3/3$ vs $n^3/2$), for a similar numerical stability.

Anyway, Gauss-Jordan does not require a backsubstitution step, which makes the code a little more compact. On the other hand, the reduced form is not compatible with LU decomposition.

From a theoretical point of view, the reduced form has the advantage of being uniquely defined for a given matrix.

Anyway, it is worth to note that for an invertible matrix, the reduced form is simply a unit matrix, totally uniniteresting. The concept is only useful for degenerate cases.

What is the use of reduced row echelon form (not a row echelon form)?

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