Cmloegcmluin at 04:43, 31 December 2021

2021-12-31T04:43:17Z

← Older revision		Revision as of 04:43, 31 December 2021
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	- [[User:Sintel\|Sintel]] ([[User talk:Sintel\|talk]]) 22:11, 30 December 2021 (UTC)		- [[User:Sintel\|Sintel]] ([[User talk: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. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 04:43, 31 December 2021 (UTC)

Sintel: some notes

2021-12-30T22:11:20Z

some notes

New page

== 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 [[transversal generators|"transversals"]] for example.)

- [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 22:11, 30 December 2021 (UTC)

Talk:Matrix echelon forms - Revision history

Cmloegcmluin at 04:43, 31 December 2021

Sintel: some notes