Defactoring algorithms: Difference between revisions

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{{Todo|inline=1| rework |text=Try explaining without wedgies outside historical notes.}}

This article discusses how to identify [[enfactoring]] in [[regular temperament]] [[mapping]]s and then [[defactoring|defactor]] it.

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<math>

\left[ \begin{array} {ccc|cc}

0 & -11 & 4 & 13 & -6

0 & -11 & 4 & 13 & -6 \\

2 & 5 & 4 & -2 & 1 \\

\end{array} \right]

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\end{matrix} \right]</math>

The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref group="note">In the doubly-enfactored case of {{rket|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of {{nowrap|33 {{=}} 3 × 11}}, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref group="note">It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with &~~inus~~;3 of the 2nd row, i.e. 2{{map|6 5 -4}} + (−3){{map|4 -4 1}} {{=}} {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{rbra|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: (−1){{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} {{=}} {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.

The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref group="note">In the doubly-enfactored case of {{rket|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of {{nowrap|33 {{=}} 3 × 11}}, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref group="note">It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with −3 of the 2nd row, i.e. 2{{map|6 5 -4}} + (−3){{map|4 -4 1}} {{=}} {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{rbra|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: (−1){{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} {{=}} {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.

}}

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[[Category:Math]]

[[Category:Pages with proofs]]

[[Category:Algorithms]]

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+{{Todo|inline=1| rework |text=Try explaining without wedgies outside historical notes.}}
 This article discusses how to identify [[enfactoring]] in [[regular temperament]] [[mapping]]s and then [[defactoring|defactor]] it.
@@ Line 339: / Line 341: @@
 <math>
 \left[ \begin{array} {ccc|cc}
-& -11 & 4  &  13 & -6
+& -11 & 4  &  13 & -6 \\
 & 5 & 4  &  -2 & 1 \\
 \end{array} \right]
@@ Line 782: / Line 784: @@
 \end{matrix} \right]</math>
-The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref group="note">In the doubly-enfactored case of {{rket|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of {{nowrap|33 {{=}} 3 &#215; 11}}, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref group="note">It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with &inus;3 of the 2nd row, i.e. 2{{map|6 5 -4}} + (&minus;3){{map|4 -4 1}} {{=}} {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{rbra|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: (&minus;1){{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} {{=}} {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.
+The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref group="note">In the doubly-enfactored case of {{rket|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of {{nowrap|33 {{=}} 3 &#215; 11}}, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref group="note">It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with &minus;3 of the 2nd row, i.e. 2{{map|6 5 -4}} + (&minus;3){{map|4 -4 1}} {{=}} {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{rbra|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: (&minus;1){{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} {{=}} {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.
 }}
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 [[Category:Math]]
 [[Category:Pages with proofs]]
+[[Category:Algorithms]]