Matrix echelon forms: Difference between revisions

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So HNF has a lot in common with IRREF, which is the IREF you find by converting the RREF, but it is not always the same as IRREF.

[[~~Canonical~~ form]] is closely related to HNF, because the second step of finding the ~~DCF~~ is taking the HNF. So the canonical form is always ''a'' HNF, and therefore it has all the same properties of being echelon, integer, and normalized, and in turn therefore it also provides a unique representation. However it is not necessary ''the'' same HNF of the original mapping, due to the first step being defactoring; it is the same as as HNF except when the original mapping is enfactored.

[[Defactored Hermite form]] (which we will call "canonical form" here) is closely related to HNF, because the second step of finding the canonical form is taking the HNF. So the canonical form is always ''a'' HNF, and therefore it has all the same properties of being echelon, integer, and normalized, and in turn therefore it also provides a unique representation. However it is not necessary ''the'' same HNF of the original mapping, due to the first step being defactoring; it is the same as as HNF except when the original mapping is enfactored.

In the below example, <math>p_{ij}</math> represents any positive integer, and <math>a_{ij}</math> represents any nonnegative integer less than the <math>p</math> in its column.

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

Considering only full-rank, integer mappings, we find three cases for a given temperament which is not enfactored. In all three cases, HNF is the same as canonical form (here abbreviated as DCF, for defactored canonical form):

Considering only full-rank, integer mappings, we find three cases for a given temperament which is not enfactored. In all three cases, HNF is the same as canonical form (here abbreviated as DCF, for defactored canonical form<ref>This was before the community decided on "defactored Hermite form", and I was too lazy to go back and update all these diagrams.</ref>):

# The RREF, IRREF, and HNF are all ''different''. Example: [[Porcupine_family#Porcupine|porcupine]] with RREF of {{ket|{{map|1 0 <math>-\frac13</math>}} {{map|0 1 <math>\frac53</math>}}}}, IRREF of {{ket|{{map|3 0 -1}} {{map|0 3 5}}}}, and HNF of {{ket|{{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 {{ket|{{map|1 0 -4}} {{map|0 1 4}}}}. This case is quite rare.

@@ Line 142: / Line 142: @@
 So HNF has a lot in common with IRREF, which is the IREF you find by converting the RREF, but it is not always the same as IRREF.
-[[Canonical form]] is closely related to HNF, because the second step of finding the DCF is taking the HNF. So the canonical form is always ''a'' HNF, and therefore it has all the same properties of being echelon, integer, and normalized, and in turn therefore it also provides a unique representation. However it is not necessary ''the'' same HNF of the original mapping, due to the first step being defactoring; it is the same as as HNF except when the original mapping is enfactored.
+[[Defactored Hermite form]] (which we will call "canonical form" here) is closely related to HNF, because the second step of finding the canonical form is taking the HNF. So the canonical form is always ''a'' HNF, and therefore it has all the same properties of being echelon, integer, and normalized, and in turn therefore it also provides a unique representation. However it is not necessary ''the'' same HNF of the original mapping, due to the first step being defactoring; it is the same as as HNF except when the original mapping is enfactored.
 In the below example, <span><math>p_{ij}</math></span> represents any positive integer, and <span><math>a_{ij}</math></span> represents any nonnegative integer less than the <span><math>p</math></span> in its column.
@@ Line 197: / Line 197: @@
 [[File:Cases for temperament mapping forms3.png|300px|thumb|right]]
-Considering only full-rank, integer mappings, we find three cases for a given temperament which is not enfactored. In all three cases, HNF is the same as canonical form (here abbreviated as DCF, for defactored canonical form):
+Considering only full-rank, integer mappings, we find three cases for a given temperament which is not enfactored. In all three cases, HNF is the same as canonical form (here abbreviated as DCF, for defactored canonical form<ref>This was before the community decided on "defactored Hermite form", and I was too lazy to go back and update all these diagrams.</ref>):
 # The RREF, IRREF, and HNF are all ''different''. Example: [[Porcupine_family#Porcupine|porcupine]] with RREF of {{ket|{{map|1 0 <span><math>-\frac13</math></span>}} {{map|0 1 <span><math>\frac53</math></span>}}}}, IRREF of {{ket|{{map|3 0 -1}} {{map|0 3 5}}}}, and HNF of {{ket|{{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 {{ket|{{map|1 0 -4}} {{map|0 1 4}}}}. This case is quite rare.