Temperament addition: Difference between revisions

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It turns out that you can always isolate the enfactoring factor in the single final vector of the matrix — the linearly independent vector — through linear combinations of the vectors in the <math>L_{\text{dep}}</math>. In this case, since there's only a single vector in the <math>L_{\text{dep}}</math>, therefore all we need to do is repeatedly add that one linearly dependent vector to the linearly independent vector until we find a vector with the target GCD, which we can then simply divide out to defactor the matrix.

In this case, we can accomplish this by adding 11 times the first vector. For the first matrix, {{map|1 0 -4 -13}} + 11⋅{{map|19 30 44 53}}={{map|210 330 480 570}}, whose entries have a GCD=30, so we can defactor the matrix by dividing that vector by 30, leaving us with {{map|7 11 16 19}}. Therefore the final matrix here is [{{map|19 30 44 53}} {{map|7 11 16 19}}⟩. The other matrix matrix happens to defactor in the same way: {{map|1 0 -4 17}} + 11⋅{{map|19 30 44 53}}={{map|210 330 480 600}} whose GCD is also 30, reducing to {{map|7 11 16 20}}, so the final matrix is [{{map|19 30 44 53}} {{map|7 11 16 20}}⟩.

In this case, we can accomplish this by adding 11 times the first vector. For the first matrix, {{map|1 0 -4 -13}} + 11⋅{{map|19 30 44 53}} = {{map|210 330 480 570}}, whose entries have a GCD = 30, so we can defactor the matrix by dividing that vector by 30, leaving us with {{map|7 11 16 19}}. Therefore the final matrix here is [{{map|19 30 44 53}} {{map|7 11 16 19}}⟩. The other matrix matrix happens to defactor in the same way: {{map|1 0 -4 17}} + 11⋅{{map|19 30 44 53}} = {{map|210 330 480 600}} whose GCD is also 30, reducing to {{map|7 11 16 20}}, so the final matrix is [{{map|19 30 44 53}} {{map|7 11 16 20}}⟩.

Now the matrices are ready to add:

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 It turns out that you can always isolate the enfactoring factor in the single final vector of the matrix — the <span style="color: #B6321C;">linearly independent vector</span> — through linear combinations of the vectors in the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span>. In this case, since there's only a single vector in the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span>, therefore all we need to do is repeatedly add that <span style="color: #3C8031;">one linearly dependent vector</span> to the <span style="color: #B6321C;">linearly independent vector</span> until we find a vector with the target GCD, which we can then simply divide out to defactor the matrix.
-In this case, we can accomplish this by adding 11 times the first vector. For the first matrix, {{map|1 0 -4 -13}} + 11⋅<span style="color: #3C8031;">{{map|19 30 44 53}}</span>={{map|210 330 480 570}}, whose entries have a GCD=30, so we can defactor the matrix by dividing that vector by 30, leaving us with <span style="color: #B6321C;">{{map|7 11 16 19}}</span>. Therefore the final matrix here is [<span style="color: #3C8031;">{{map|19 30 44 53}}</span> {{map|7 11 16 19}}⟩. The other matrix matrix happens to defactor in the same way: {{map|1 0 -4 17}} + 11⋅<span style="color: #3C8031;">{{map|19 30 44 53}}</span>={{map|210 330 480 600}} whose GCD is also 30, reducing to <span style="color: #B6321C;">{{map|7 11 16 20}}</span>, so the final matrix is [<span style="color: #3C8031;">{{map|19 30 44 53}}</span> <span style="color: #B6321C;">{{map|7 11 16 20}}</span>⟩.
+In this case, we can accomplish this by adding 11 times the first vector. For the first matrix, {{map|1 0 -4 -13}} + 11⋅<span style="color: #3C8031;">{{map|19 30 44 53}}</span> = {{map|210 330 480 570}}, whose entries have a GCD = 30, so we can defactor the matrix by dividing that vector by 30, leaving us with <span style="color: #B6321C;">{{map|7 11 16 19}}</span>. Therefore the final matrix here is [<span style="color: #3C8031;">{{map|19 30 44 53}}</span> {{map|7 11 16 19}}⟩. The other matrix matrix happens to defactor in the same way: {{map|1 0 -4 17}} + 11⋅<span style="color: #3C8031;">{{map|19 30 44 53}}</span> = {{map|210 330 480 600}} whose GCD is also 30, reducing to <span style="color: #B6321C;">{{map|7 11 16 20}}</span>, so the final matrix is [<span style="color: #3C8031;">{{map|19 30 44 53}}</span> <span style="color: #B6321C;">{{map|7 11 16 20}}</span>⟩.
 Now the matrices are ready to add: