Defactoring: Difference between revisions

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The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref>In the doubly-enfactored case of {{vector|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of 33 = 3 × 11, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref>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 {{vector|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{map|{{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.

== other defactoring methods ==

When in development on an ideal defactoring method — the effort which culminated in column Hermite defactoring — Dave and Douglas experimented on other methods:

* [[Sum-and-difference_defactoring|SAD defactoring]]

* [[MADAM defactoring]]

= canonical comma-bases =

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 The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref>In the doubly-enfactored case of {{vector|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of 33 = 3 × 11, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref>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 {{vector|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{map|{{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.
+== other defactoring methods ==
+When in development on an ideal defactoring method — the effort which culminated in column Hermite defactoring — Dave and Douglas experimented on other methods:
+* [[Sum-and-difference_defactoring|SAD defactoring]]
+* [[MADAM defactoring]]
 = canonical comma-bases =