Matrix echelon forms: Difference between revisions

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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>

In the below example, ~~~~<math>x_{ij}</math~~></span~~> represents any number.

In the below example, <math>x_{ij}</math> represents any number.

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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.

In the below example, ~~~~<math>n_{ij}</math~~></span~~> represents any integer.

In the below example, <math>n_{ij}</math> represents any integer.

{| class="wikitable" style="text-align: center;"

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[[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~~></span~~> represents any positive integer, and ~~~~<math>a_{ij}</math~~></span~~> represents any nonnegative integer less than the ~~~~<math>p</math~~></span~~> in its column.

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.

{| class="wikitable" style="text-align: center;"

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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<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~~></span~~>}} {{map|0 1 ~~~~<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, 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.

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

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

And there are three corresponding cases when a temperament is enfactored. In all three cases, the key difference is that HNF is no longer the same as DCF, with the only difference being that the common factor is not removed. In all cases below, the examples are shown with a common factor of 2 introduced in their second row, which stays behind in the second row of the HNF:

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# 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. {{ket|{{map|15 24 35}} {{map|38 60 88}}}} causes the HNF to be {{ket|{{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 {{ket|{{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 {{ket|{{map|33 0 -131}} {{map|0 33 131}}}}, DCF of {{ket|{{map|1 1 0}} {{map|0 33 131}}}}, and RREF of {{ket|{{map|1 0 ~~~~<math>\frac{-131}{33}</math~~></span~~>}} {{map|0 1 ~~~~<math>\frac{131}{33}</math~~></span~~>}}}}.

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 {{ket|{{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 {{ket|{{map|33 0 -131}} {{map|0 33 131}}}}, DCF of {{ket|{{map|1 1 0}} {{map|0 33 131}}}}, and RREF of {{ket|{{map|1 0 <math>\frac{-131}{33}</math>}} {{map|0 1 <math>\frac{131}{33}</math>}}}}.

That accounts for 7 of the 15 total possible cases for a system of equalities between 4 entities. The remaining 9 cases are impossible due to properties of the domain:

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 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>
-In the below example, <span><math>x_{ij}</math></span> represents any number.
+In the below example, <math>x_{ij}</math> represents any number.
 {| class="wikitable" style="text-align: center;"
@@ Line 32: / Line 32: @@
 '''[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.
-In the below example, <span><math>n_{ij}</math></span> represents any integer.
+In the below example, <math>n_{ij}</math> represents any integer.
 {| class="wikitable" style="text-align: center;"
@@ Line 144: / Line 144: @@
 [[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.
+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.
 {| class="wikitable" style="text-align: center;"
@@ Line 198: / Line 198: @@
 [[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<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, 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.
-# The IRREF and HNF are the same, but the ''RREF is different''. Example: [[Kleismic_family#Hanson|hanson]] with IRREF and HNF of {{ket|{{map|1 0 1}} {{map|0 6 5}}}} but RREF of {{ket|{{map|1 0 1}} {{map|0 1 <span><math>\frac56</math></span>}}}}.
+# The IRREF and HNF are the same, but the ''RREF is different''. Example: [[Kleismic_family#Hanson|hanson]] with IRREF and HNF of {{ket|{{map|1 0 1}} {{map|0 6 5}}}} but RREF of {{ket|{{map|1 0 1}} {{map|0 1 <math>\frac56</math>}}}}.
 And there are three corresponding cases when a temperament is enfactored. In all three cases, the key difference is that HNF is no longer the same as DCF, with the only difference being that the common factor is not removed. In all cases below, the examples are shown with a common factor of 2 introduced in their second row, which stays behind in the second row of the HNF:
@@ Line 207: / Line 207: @@
 # 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. {{ket|{{map|15 24 35}} {{map|38 60 88}}}} causes the HNF to be {{ket|{{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 {{ket|{{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 {{ket|{{map|33 0 -131}} {{map|0 33 131}}}}, DCF of {{ket|{{map|1 1 0}} {{map|0 33 131}}}}, and RREF of {{ket|{{map|1 0 <span><math>\frac{-131}{33}</math></span>}} {{map|0 1 <span><math>\frac{131}{33}</math></span>}}}}.
+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 {{ket|{{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 {{ket|{{map|33 0 -131}} {{map|0 33 131}}}}, DCF of {{ket|{{map|1 1 0}} {{map|0 33 131}}}}, and RREF of {{ket|{{map|1 0 <math>\frac{-131}{33}</math>}} {{map|0 1 <math>\frac{131}{33}</math>}}}}.
 That accounts for 7 of the 15 total possible cases for a system of equalities between 4 entities. The remaining 9 cases are impossible due to properties of the domain: