Matrix echelon forms: Difference between revisions

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: {{vector|{{map|1 0 0}} {{map|0 1 0}}}}. Or, if you 2-enfactor them, they will all have the SNF {{vector|{{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>

# The RREF, IRREF, and HNF are all ''different''. Example: [[Porcupine_family#Porcupine|porcupine]] with RREF of {{vector|{{map|1 0 <span><math>-\frac13</math></span>}} {{map|0 1 <span><math>\frac53</math></span>}}}}, IRREF of {{vector|{{map|3 0 -1}} {{map|0 3 5}}}}, and HNF of {{vector|{{map|1 2 3}} {{map|0 3 5}}}}.  

# The RREF, IRREF, HNF are all ''the same''. Example: [[Meantone_family#Meantone_.2812.2619.2C_2.3.5.29|meantone]] with all equal to {{vector|{{map|1 0 -4}} {{map|0 1 4}}}}. This case is quite rare.

# The IRREF and HNF are the same, but the ''RREF is different''. Example: [[Kleismic_family#Hanson|hanson]] with IRREF and HNF of {{vector|{{map|1 0 1}} {{map|0 6 5}}}} but RREF of {{vector|{{map|1 0 1}} {{map|0 1 <span><math>\frac56</math></span>}}}}.

# Now ''all four'' are different. Example: enfactored porcupine, e.g. {{vector|{{map|15 24 35}} {{map|14 22 32}}}} causes the HNF to be {{vector|{{map|1 2 3}} {{map|0 6 10}}}}.

# Everything is still the same now, ''except HNF''. Example: enfactored meantone, e.g. {{vector|{{map|5 8 12}} {{map|14 22 32}}}} causes the HNF to be {{vector|{{map|1 0 -4}} {{map|0 2 8}}}}. This case, like the corresponding defactored state, is also quite rare.

# The ''only match'' now is between IRREF and DCF. In other words, the HNF and DCF diverged, and it was the DCF which remained the same as IRREF. Example: enfactored hanson, e.g. {{vector|{{map|15 24 35}} {{map|38 60 88}}}} causes the HNF to be {{vector|{{map|1 0 1}} {{map|0 12 10}}}}.

There is also a final case which is incredibly rare. It can be compared to the #3 cases above, the ones using hanson as their example. The idea here is that when the HNF and DCF diverge, instead of DCF remaining the same as IRREF, it's the HNF that remains the same as IRREF. There may be no practical temperoids with this case, but {{vector|{{map|165 264 393}} {{map|231 363 524}}}} will do it<ref>AKA 165b⁴c¹⁹&231b⁶c²⁴, which tempers out the 7.753¢ comma {{vector|-131 131 -33}}!</ref>, with IRREF and HNF of {{vector|{{map|33 0 -131}} {{map|0 33 131}}}}, DCF of {{vector|{{map|1 1 0}} {{map|0 33 131}}}}, and RREF of {{vector|{{map|1 0 <span><math>\frac{-131}{33}</math></span>}} {{map|0 1 <span><math>\frac{131}{33}</math></span>}}}}.

* The case where the only equality is between RREF and IRREF would only be possible when the temperament is both enfactored and rank-deficient, such as {{vector|{{map|0 0 0}} {{map|2 4 6}}}} which gives to HNF, RREF, IRREF, and DCF of {{vector|{{map|2 4 6}} {{map|0 0 0}}}}, {{vector|{{map|1 2 3}} {{map|0 0 0}}}}, {{vector|{{map|1 2 3}} {{map|0 0 0}}}}, and {{vector|{{map|1 2 3}}}}, respectively.