Talk:Matrix echelon forms

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Some notes

It should be noted that over the integer ring ([math]\mathbb{Z}[/math]), the only actual equivalent of the echelon form is the HNF. RREF is also useful when you extend your ring to a field [math]\mathbb{Q}[/math] or [math]\mathbb{R}[/math]. (Which is sometimes done in the context of generator optimization or deriving projection matrices.) I don't really see the point of the other ones listed here.

Also, these forms can be thought of more generally as a decompositions, into two or more matrices. For example the HNF of a gives A = U*T, where T is in the HNF form of A and U is unimodular. This view also shows the usefulness of the SNF. The SNF itself seems useless but its actually a decomposition into A = L*D*R, where D is diagonal and L, R unimodular. This is extremely useful when trying to solve systems of equations in [math]\mathbb{Z}[/math]. (You can use it to calculate "transversals" for example.)

- Sintel (talk) 22:11, 30 December 2021 (UTC)

Thanks as always for your critical feedback, Sintel. I think these could be valuable additions to the article itself if you want to incorporate them. I note that the U of the Hermite decomposition is used in Pernet-Stein and column Hermite defactoring, and the R of the SNF decomposition is used in Smith defactoring. These are discussed on the defactoring algorithms page. --Cmloegcmluin (talk) 04:43, 31 December 2021 (UTC)