Matrix echelon forms: Difference between revisions

The mathematical field of linear algebra has defined a number of echelon forms for matrices, many of them have been used in [[regular temperament theory]]. This article is intended as a reference and gathering place for relevant information about these matrix forms.

The most general form, with the fewest constraints, is simply called '''[https://en.wikipedia.org/wiki/Row_echelon_form Row Echelon Form]''', or '''REF'''. Its only constraint is ''echelon<ref>The name "echelon" is a French word for a military troop formation with a similar triangular shape: https://en.wikipedia.org/wiki/Echelon_formation.</ref> form'': each row's pivot, or first nonzero entry, is strictly to the right of the preceding row's pivot. This single constraint is fairly weak, and therefore REF does not produce a unique representation. This constraint is shared by every matrix form discussed here.<ref>Note that the definition of REF is inconsistent and sometimes it includes some of the constraints of RREF, discussed further below. See: https://www.statisticshowto.com/matrices-and-matrix-algebra/reduced-row-echelon-form-2/</ref><ref>REF also requires that all rows that are entirely zeros should appear at the bottom of the matrix. However this rule is only relevant for [[rank-deficient]] matrices. We'll be assuming all matrices here are [[full-rank]], so we don't have to worry about this.</ref>

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'''[https://people.sc.fsu.edu/~jburkardt/f_src/row_echelon_integer/row_echelon_integer.html Integer Row Echelon Form]''', or '''IREF''', is, unsurprisingly, any REF which meets an additional ''integer'' constraint, or in other words, that all of its entries are integers. This is still not a sufficiently strict set of constraints to ensure a unique representation.

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'''[https://en.wikipedia.org/wiki/Row_echelon_form#Reduced_row_echelon_form Reduced Row Echelon Form]''', or '''RREF''', takes REF in a different direction than IREF. Instead of stipulating anything about integers, it requires that the matrix is ''reduced'', i.e. that the pivots are exactly equal to 1. This may require dividing rows by a number such that some resulting entries are no longer integers. Actually, there's a second part to the "reduced" constraint: each pivot column (a column which contains any row's pivot) has zeros for all other entries besides the pivot it contains. Due to these strict constraints, the RREF of a matrix is the first one we've looked at so far here which does ensure uniqueness.

So IREF and RREF make a [https://en.wikipedia.org/wiki/Venn_diagram 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).

'''[[Normal_lists|Integer Reduced Row Echelon Form]]''', or '''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 [https://en.wikipedia.org/wiki/Euler_diagram Euler diagram] (generalization of Venn diagram) would only exist within their intersection. However this is not the case. That's 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.<ref>Alternatively, IRREF can be computed by finding the nullspace of a mapping, or in other words, the corresponding [[comma basis]] for the temperament represented by the mapping, and then in turn taking the nullspace of the comma basis to get back to an equivalent mapping. The relationship between the process for finding the IRREF from the RREF and this process is not proven, but thousands of automated tests run as an experiment strongly suggest that these two techniques are equivalent.<br>

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

'''[https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form]''', or '''HNF''': this one's constraints begin with echelon form and integer, therefore every HNF is also IREF. But HNF is not exactly reduced (as discussed above; [[canonical_form#RREF|link here]]); instead, it is ''normalized'', which — similarly to reduced — is a two-part constraint. Where reduced requires that all pivots be exactly equal to 1, normalized requires only that all pivots be positive (positive integers, of course, due to the other integer constraint). And where reduced requires that all entries in pivot columns besides the pivots are exactly equal to 0, normalized requires only that all entries in pivot columns below the pivots are exactly equal to 0, while entries in pivot columns above the pivots only have to be strictly less than the pivot in the respective column (while still being non-negative).<ref>The exact criteria for HNF are not always consistently agreed upon, however.</ref><ref>We are using "row-style" Hermite Normal Form here, not "column-style"; the latter would involve simply flipping everything 90 degrees so that the echelon requirement was that pivots be strictly ''below'' the pivots in the previous ''column'', and that pivot ''rows'' are considered for the normalization constraint rather than pivot ''columns''.</ref> In other words, elements above the pivot have to be reduced modulo the pivot. The normalization HNF uses is cool because this constraint, while strictly less strict than the reduced constraint used by RREF, is still strict enough to ensure uniqueness, but loose enough to ensure the integer constraint can be simultaneously satisfied, where RREF cannot ensure that.  

For an example of working out a HNF by hand, see: [[defactoring#Hermite decomposition by hand]].

There is also the '''[https://en.wikipedia.org/wiki/Smith_normal_form Smith Normal Form]''', or '''SNF''', but we won't be discussing it in this context, because putting a mapping into SNF obliterates a lot of meaningful RTT information. SNF is also echelon, and integer, so like HNF it is also always IREF. But SNF requires that every single entry other than the pivots are zero, and that the pivots all fall exactly along the main diagonal of the matrix. The SNF essentially reduces a matrix down to the information of what its rank is and whether it is enfactored. For example, all 5-limit rank-2 temperaments such as meantone, porcupine, mavila, hanson, etc. have the same SNF: {{rket|{{map|1 0 0}} {{map|0 1 0}}}}. Or, if you 2-enfactor them, they will all have the SNF {{rket|{{map|1 0 0}} {{map|0 2 0}}}}. So while the SNF is closely related to defactoring, it is not itself a useful form to put mappings into.<ref>Here is a useful resource for computing the Smith Normal Form manually, if you are interested: https://math.stackexchange.com/questions/133076/computing-the-smith-normal-form The fact that it involves calculating so many GCDs is unsurprising given its ability to defactor matrices.</ref>

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[[File:Cases for temperament mapping forms3.png|300px|thumb|right]]

* The case where the only equalities are between RREF and IRREF and between HNF and DCF is impossible, because if RREF=IRREF that suggests that all entries are multiples of their pivots, which is easy if the temperament is enfactored, but if HNF=DCF then it is not.

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