Matrix echelon forms: Difference between revisions

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{{Wikipedia|Row echelon form #Reduced row echelon form}}

So IREF and RREF make a {{w|Venn diagram}} inside the category of REF: some IREF are RREF, but there are some RREF that are not IREF and some IREF that are not RREF. When we scope the situation to a specific matrix, however, because RREF is a unique form, this means that one or the other sector of the Venn diagram for RREF will be empty; either the unique RREF will also be IREF (and therefore the RREF-but-not-IREF sector will be empty), or it will not be IREF (and vice versa).

'''Integer reduced row echelon form''' ('''IRREF'''): based on the name, one might expect this form to be a combination of the constraints for RREF and IREF, and therefore if represented in an {{w|Euler diagram}} (generalization of Venn diagram) would only exist within their intersection. However this is not the case. That is because the IRREF does not include the key constraint of RREF which is that all of the pivots must be 1. IRREF is produced by simply taking the unique RREF and multiplying each row by whatever minimum value is necessary to make all of the entries integers. Of course, this sometimes results in the pivots no longer being 1, so sometimes it is no longer RREF. It is always still REF, though,<ref group="note">Also, it will always still satisfy the second aspect of reduced, i.e. that all other entries in pivot columns besides the pivots are zeroes.</ref> and because it is also always integer, that makes it always IREF; therefore, IRREF is strictly a subcategory of IREF. And because the RREF is unique, and the conversion process does not alter that, the IRREF is also unique.   

It is not possible for an RREF to be IREF without also being IRREF.  

{{Wikipedia|Hermite normal form}}

{{Main|Normal forms #Hermite normal form}}