Defactoring: Difference between revisions

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</math>

Note that the interval vectors are columns when put into matrix form like this. So now we antitranspose, or in other words, transpose the matrix but instead of across its main diagonal (top-left to bottom-right) as with the traditional transpose, across its antidiagonal (top-right to bottom-left). This has the effect of both reversing the entries within each interval, as well as reversing the order of the intervals themselves.<ref>Because these are going to be put into HNF soon, the reversing of the order of the intervals themselves is irrelevant. But it is important that the order of the intervals themselves reverses on the way out, in the second antitranspose. And so for simplicity of explanation's sake, we simply say to do an antitranspose at both the beginning and end of the operation.</ref>

Note that the interval vectors are columns when put into matrix form like this.

So now we antitranspose, or in other words, transpose the matrix but instead of across its main diagonal (top-left to bottom-right) as with the traditional transpose, across its antidiagonal (top-right to bottom-left).

<math>

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\end{array}

</math>

This has the effect of both reversing the entries within each interval, as well as reversing the order of the intervals themselves. The purpose of these reversals is so that when the HNF tries to put all the zeros in the bottom-left corner, it gravitates them toward where we want them: the higher primes, and commas that will be earlier in the list after the second antitranspose<ref>Because these are going to be put into HNF soon, the reversing of the order of the intervals themselves at the beginning is irrelevant. But it is important that the order of the intervals themselves reverses on the way out, in the second antitranspose. And so for simplicity of explanation's sake, we simply say to do an antitranspose at both the beginning and end of the operation.</ref>.

Now we can defactor and HNF this as if it were a mapping.

@@ Line 882: / Line 882: @@
 </math>
-Note that the interval vectors are columns when put into matrix form like this. So now we antitranspose, or in other words, transpose the matrix but instead of across its main diagonal (top-left to bottom-right) as with the traditional transpose, across its antidiagonal (top-right to bottom-left). This has the effect of both reversing the entries within each interval, as well as reversing the order of the intervals themselves.<ref>Because these are going to be put into HNF soon, the reversing of the order of the intervals themselves is irrelevant. But it is important that the order of the intervals themselves reverses on the way out, in the second antitranspose. And so for simplicity of explanation's sake, we simply say to do an antitranspose at both the beginning and end of the operation.</ref>
+Note that the interval vectors are columns when put into matrix form like this.
+So now we antitranspose, or in other words, transpose the matrix but instead of across its main diagonal (top-left to bottom-right) as with the traditional transpose, across its antidiagonal (top-right to bottom-left).
 <math>
@@ Line 903: / Line 905: @@
 \end{array}
 </math>
+This has the effect of both reversing the entries within each interval, as well as reversing the order of the intervals themselves. The purpose of these reversals is so that when the HNF tries to put all the zeros in the bottom-left corner, it gravitates them toward where we want them: the higher primes, and commas that will be earlier in the list after the second antitranspose<ref>Because these are going to be put into HNF soon, the reversing of the order of the intervals themselves at the beginning is irrelevant. But it is important that the order of the intervals themselves reverses on the way out, in the second antitranspose. And so for simplicity of explanation's sake, we simply say to do an antitranspose at both the beginning and end of the operation.</ref>.
 Now we can defactor and HNF this as if it were a mapping.