Defactoring: Difference between revisions

Line 260:

smithDefactor[m_] := Take[Inverse[rightReducingMatrix[m]], MatrixRank[m]]</nowiki>

So the first thing that happens to <math>m</math> when you pass it in to <code>smithDefactor[]</code> is that it calls <code>rightReducingMatrix[]</code> on it. This will find the Smith decomposition (using a function built in to Wolfram Language), which gives you three outputs: the Smith normal form, flanked by its left and right reducing matrices. ~~We're asked~~ only for the right reducing matrix, so we grab that with <code>Last[]</code>. So that's what the function on the first line, <code>rightReducingMatrix[]</code>, does.

So the first thing that happens to <math>m</math> when you pass it in to <code>smithDefactor[]</code> is that it calls <code>rightReducingMatrix[]</code> on it. This will find the Smith decomposition (using a function built in to Wolfram Language), which gives you three outputs: the Smith normal form, flanked by its left and right reducing matrices. Gene asks only for the right reducing matrix, so we grab that with <code>Last[]</code>. So that's what the function on the first line, <code>rightReducingMatrix[]</code>, does.

Then Gene asks us to invert this result and take its first <math>r</math> rows, where <math>r</math> is the rank of the temperament. <code>Invert[]</code> takes care of the inversion, of course. <code>MatrixRank[m]</code> gives the count of linearly independent rows to the mapping, AKA the rank, or count of generators in this temperament. In this case that's 2. And so <code>Take[list, 2]</code> simply returns the first 2 entries of the list.

Almost done! Except Gene not only defactors, he also calls for HNF, as we would, to achieve canonical form.

Almost done! Except Gene not only defactors, he also calls for HNF, as we would, to achieve canonical (unique ID) form.

<nowiki>normalize[m_] := Last[HermiteDecomposition[m]]</nowiki>

@@ Line 260: / Line 260: @@
 smithDefactor[m_] := Take[Inverse[rightReducingMatrix[m]], MatrixRank[m]]</nowiki>
-So the first thing that happens to <span><math>m</math></span> when you pass it in to <code>smithDefactor[]</code> is that it calls <code>rightReducingMatrix[]</code> on it. This will find the Smith decomposition (using a function built in to Wolfram Language), which gives you three outputs: the Smith normal form, flanked by its left and right reducing matrices. We're asked only for the right reducing matrix, so we grab that with <code>Last[]</code>. So that's what the function on the first line, <code>rightReducingMatrix[]</code>, does.
+So the first thing that happens to <span><math>m</math></span> when you pass it in to <code>smithDefactor[]</code> is that it calls <code>rightReducingMatrix[]</code> on it. This will find the Smith decomposition (using a function built in to Wolfram Language), which gives you three outputs: the Smith normal form, flanked by its left and right reducing matrices. Gene asks only for the right reducing matrix, so we grab that with <code>Last[]</code>. So that's what the function on the first line, <code>rightReducingMatrix[]</code>, does.
 Then Gene asks us to invert this result and take its first <span><math>r</math></span> rows, where <span><math>r</math></span> is the rank of the temperament. <code>Invert[]</code> takes care of the inversion, of course. <code>MatrixRank[m]</code> gives the count of linearly independent rows to the mapping, AKA the rank, or count of generators in this temperament. In this case that's 2. And so <code>Take[list, 2]</code> simply returns the first 2 entries of the list.
-Almost done! Except Gene not only defactors, he also calls for HNF, as we would, to achieve canonical form.
+Almost done! Except Gene not only defactors, he also calls for HNF, as we would, to achieve canonical (unique ID) form.
   <nowiki>normalize[m_] := Last[HermiteDecomposition[m]]</nowiki>