Defactoring: Difference between revisions
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Sometimes the enfactoring of a mapping is immediately apparent. For example: | Sometimes the enfactoring of a mapping is immediately apparent. For example: | ||
{{ | <math> | ||
\left[ | |||
\begin{array} {r} | |||
24 & 38 & 56 \\ | |||
\end{array} | |||
\right] | |||
</math> | |||
This mapping has only a single row, and we can see that every element in that row is even. Therefore we have a common factor of at least 2. In this case it is in fact exactly 2. So we can say that this is a 2-enfactored mapping. | This mapping has only a single row, and we can see that every element in that row is even. Therefore we have a common factor of at least 2. In this case it is in fact exactly 2. So we can say that this is a 2-enfactored mapping. | ||
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Other times, enfactoring is less apparent. Consider this example: | Other times, enfactoring is less apparent. Consider this example: | ||
{{ | <math> | ||
\left[ | |||
\begin{array} {r} | |||
3 & 0 & -1 \\ | |||
0 & 3 & 5 \\ | |||
\end{array} | |||
\right] | |||
</math> | |||
This is a form of 5-limit [[porcupine]], a [[rank-2]] temperament. Looking at either row, neither map has a common factor. But remember that we also need to check linear combinations of rows. If we subtract the 2nd row from the 1st row, we can produce the row {{map|3 -3 -6}}, which has a common factor of 3. So this mapping is also enfactored, even though it's not obvious from just looking at it. | This is a form of 5-limit [[porcupine]], a [[rank-2]] temperament. Looking at either row, neither map has a common factor. But remember that we also need to check linear combinations of rows. If we subtract the 2nd row from the 1st row, we can produce the row {{map|3 -3 -6}}, which has a common factor of 3. So this mapping is also enfactored, even though it's not obvious from just looking at it. | ||
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=== well-hidden enfactoring === | === well-hidden enfactoring === | ||
Sometimes the hidden common factor is even harder to find. Consider the mapping {{ | Sometimes the hidden common factor is even harder to find. Consider the mapping: | ||
<math> | |||
\left[ | |||
\begin{array} {r} | |||
6 & 5 & -4 \\ | |||
4 & -4 & 1 \\ | |||
\end{array} | |||
\right] | |||
</math> | |||
To find this common factor, we need to linearly combine two of the first row {{map|6 5 -4}} and negative three of the 2nd row {{map|4 -4 1}} to produce {{map|0 22 -11}}. So we can see here that its common factor is 11. But there's no clear relationship between the numbers 2 and 3 and the number 11. And so we can begin to see that the problem of identifying enfactored mapping may not be very simple or straightforward. | |||
== defactoring methods == | == defactoring methods == | ||