Matrix echelon forms: Difference between revisions

Line 84:

== IRREF ==

'''[[Normal_lists|Integer Reduced Row Echelon Form]]''', or '''IRREF''': based on the name, one might expect this form to be a combination of the constraints for RREF and IREF, and therefore if represented in an [https://en.wikipedia.org/wiki/Euler_diagram Euler diagram] (generalization of Venn diagram) would only exist within their intersection. However this is not the case. That's because the IRREF does not include the key constraint of RREF which is that all of the pivots must be 1. IRREF is produced by simply taking the unique RREF and multiplying each row by whatever minimum value is necessary to make all of the entries integers<ref>Alternatively, IRREF can be computed by finding the nullspace of a mapping, or in other words, the corresponding [[comma basis]] for the temperament represented by the mapping, and then in turn taking the nullspace of the comma basis to get back to an equivalent mapping. The relationship between the process for finding the IRREF from the RREF and this process is not proven, but thousands of automated tests run as an experiment strongly suggest that these two techniques are equivalent.

'''[[Normal_lists|Integer Reduced Row Echelon Form]]''', or '''IRREF''': based on the name, one might expect this form to be a combination of the constraints for RREF and IREF, and therefore if represented in an [https://en.wikipedia.org/wiki/Euler_diagram Euler diagram] (generalization of Venn diagram) would only exist within their intersection. However this is not the case. That's because the IRREF does not include the key constraint of RREF which is that all of the pivots must be 1. IRREF is produced by simply taking the unique RREF and multiplying each row by whatever minimum value is necessary to make all of the entries integers.<ref>Alternatively, IRREF can be computed by finding the nullspace of a mapping, or in other words, the corresponding [[comma basis]] for the temperament represented by the mapping, and then in turn taking the nullspace of the comma basis to get back to an equivalent mapping. The relationship between the process for finding the IRREF from the RREF and this process is not proven, but thousands of automated tests run as an experiment strongly suggest that these two techniques are equivalent.

rref[m_] := RowReduce[m]

Line 109:

There is a difference in that IRREF does not remove rows of zeros in the end for rank-deficient mappings, while this "nullspace'n'back" does, but for most normal cases, they're the same.

</ref>. Of course, this sometimes results in the pivots no longer being 1, so sometimes it is no longer RREF. It is always still REF, though<ref>Also, it will always still satisfy the second aspect of reduced, i.e. that all other entries in pivot columns besides the pivots are zeroes.</ref>, and because it is also always integer, that makes it always IREF; therefore, IRREF is strictly a subcategory of IREF. And because the RREF is unique, and the conversion process does not alter that, the IRREF is also unique.

</ref> Of course, this sometimes results in the pivots no longer being 1, so sometimes it is no longer RREF. It is always still REF, though,<ref>Also, it will always still satisfy the second aspect of reduced, i.e. that all other entries in pivot columns besides the pivots are zeroes.</ref> and because it is also always integer, that makes it always IREF; therefore, IRREF is strictly a subcategory of IREF. And because the RREF is unique, and the conversion process does not alter that, the IRREF is also unique.

{| class="wikitable" style="text-align: center;"

Line 209:

# The ''only match'' now is between IRREF and DCF. In other words, the HNF and DCF diverged, and it was the DCF which remained the same as IRREF. Example: enfactored hanson, e.g. {{rket|{{map|15 24 35}} {{map|38 60 88}}}} causes the HNF to be {{rket|{{map|1 0 1}} {{map|0 12 10}}}}.

There is also a final case which is incredibly rare. It can be compared to the #3 cases above, the ones using hanson as their example. The idea here is that when the HNF and DCF diverge, instead of DCF remaining the same as IRREF, it's the HNF that remains the same as IRREF. There may be no practical temperoids with this case, but {{rket|{{map|165 264 393}} {{map|231 363 524}}}} will do it<ref>AKA 165b⁴c¹⁹&231b⁶c²⁴, which makes the 7.753¢ comma {{vector|-131 131 -33}} [[vanish]]!</ref>, with IRREF and HNF of {{rket|{{map|33 0 -131}} {{map|0 33 131}}}}, DCF of {{rket|{{map|1 1 0}} {{map|0 33 131}}}}, and RREF of {{rket|{{map|1 0 <math>\frac{-131}{33}</math>}} {{map|0 1 <math>\frac{131}{33}</math>}}}}.

There is also a final case which is incredibly rare. It can be compared to the #3 cases above, the ones using hanson as their example. The idea here is that when the HNF and DCF diverge, instead of DCF remaining the same as IRREF, it's the HNF that remains the same as IRREF. There may be no practical temperoids with this case, but {{rket|{{map|165 264 393}} {{map|231 363 524}}}} will do it,<ref>AKA 165b⁴c¹⁹&231b⁶c²⁴, which makes the 7.753¢ comma {{vector|-131 131 -33}} [[vanish]]!</ref> with IRREF and HNF of {{rket|{{map|33 0 -131}} {{map|0 33 131}}}}, DCF of {{rket|{{map|1 1 0}} {{map|0 33 131}}}}, and RREF of {{rket|{{map|1 0 <math>\frac{-131}{33}</math>}} {{map|0 1 <math>\frac{131}{33}</math>}}}}.

That accounts for 7 of the 15 total possible cases for a system of equalities between 4 entities. The remaining 9 cases are impossible due to properties of the domain:

@@ Line 84: / Line 84: @@
 == IRREF ==
-'''[[Normal_lists|Integer Reduced Row Echelon Form]]''', or '''IRREF''': based on the name, one might expect this form to be a combination of the constraints for RREF and IREF, and therefore if represented in an [https://en.wikipedia.org/wiki/Euler_diagram Euler diagram] (generalization of Venn diagram) would only exist within their intersection. However this is not the case. That's because the IRREF does not include the key constraint of RREF which is that all of the pivots must be 1. IRREF is produced by simply taking the unique RREF and multiplying each row by whatever minimum value is necessary to make all of the entries integers<ref>Alternatively, IRREF can be computed by finding the nullspace of a mapping, or in other words, the corresponding [[comma basis]] for the temperament represented by the mapping, and then in turn taking the nullspace of the comma basis to get back to an equivalent mapping. The relationship between the process for finding the IRREF from the RREF and this process is not proven, but thousands of automated tests run as an experiment strongly suggest that these two techniques are equivalent.<br>
+'''[[Normal_lists|Integer Reduced Row Echelon Form]]''', or '''IRREF''': based on the name, one might expect this form to be a combination of the constraints for RREF and IREF, and therefore if represented in an [https://en.wikipedia.org/wiki/Euler_diagram Euler diagram] (generalization of Venn diagram) would only exist within their intersection. However this is not the case. That's because the IRREF does not include the key constraint of RREF which is that all of the pivots must be 1. IRREF is produced by simply taking the unique RREF and multiplying each row by whatever minimum value is necessary to make all of the entries integers.<ref>Alternatively, IRREF can be computed by finding the nullspace of a mapping, or in other words, the corresponding [[comma basis]] for the temperament represented by the mapping, and then in turn taking the nullspace of the comma basis to get back to an equivalent mapping. The relationship between the process for finding the IRREF from the RREF and this process is not proven, but thousands of automated tests run as an experiment strongly suggest that these two techniques are equivalent.<br>
 <span style="font-family: monospace; font-size: 10px;"><br>
 rref[m_] := RowReduce[m]<br>
@@ Line 109: / Line 109: @@
 <br></span><br>
 There is a difference in that IRREF does not remove rows of zeros in the end for rank-deficient mappings, while this "nullspace'n'back" does, but for most normal cases, they're the same.
-</ref>. Of course, this sometimes results in the pivots no longer being 1, so sometimes it is no longer RREF. It is always still REF, though<ref>Also, it will always still satisfy the second aspect of reduced, i.e. that all other entries in pivot columns besides the pivots are zeroes.</ref>, and because it is also always integer, that makes it always IREF; therefore, IRREF is strictly a subcategory of IREF. And because the RREF is unique, and the conversion process does not alter that, the IRREF is also unique.
+</ref> Of course, this sometimes results in the pivots no longer being 1, so sometimes it is no longer RREF. It is always still REF, though,<ref>Also, it will always still satisfy the second aspect of reduced, i.e. that all other entries in pivot columns besides the pivots are zeroes.</ref> and because it is also always integer, that makes it always IREF; therefore, IRREF is strictly a subcategory of IREF. And because the RREF is unique, and the conversion process does not alter that, the IRREF is also unique.
 {| class="wikitable" style="text-align: center;"
@@ Line 209: / Line 209: @@
 # The ''only match'' now is between IRREF and DCF. In other words, the HNF and DCF diverged, and it was the DCF which remained the same as IRREF. Example: enfactored hanson, e.g. {{rket|{{map|15 24 35}} {{map|38 60 88}}}} causes the HNF to be {{rket|{{map|1 0 1}} {{map|0 12 10}}}}.
-There is also a final case which is incredibly rare. It can be compared to the #3 cases above, the ones using hanson as their example. The idea here is that when the HNF and DCF diverge, instead of DCF remaining the same as IRREF, it's the HNF that remains the same as IRREF. There may be no practical temperoids with this case, but {{rket|{{map|165 264 393}} {{map|231 363 524}}}} will do it<ref>AKA 165b⁴c¹⁹&231b⁶c²⁴, which makes the 7.753¢ comma {{vector|-131 131 -33}} [[vanish]]!</ref>, with IRREF and HNF of {{rket|{{map|33 0 -131}} {{map|0 33 131}}}}, DCF of {{rket|{{map|1 1 0}} {{map|0 33 131}}}}, and RREF of {{rket|{{map|1 0 <math>\frac{-131}{33}</math>}} {{map|0 1 <math>\frac{131}{33}</math>}}}}.
+There is also a final case which is incredibly rare. It can be compared to the #3 cases above, the ones using hanson as their example. The idea here is that when the HNF and DCF diverge, instead of DCF remaining the same as IRREF, it's the HNF that remains the same as IRREF. There may be no practical temperoids with this case, but {{rket|{{map|165 264 393}} {{map|231 363 524}}}} will do it,<ref>AKA 165b⁴c¹⁹&231b⁶c²⁴, which makes the 7.753¢ comma {{vector|-131 131 -33}} [[vanish]]!</ref> with IRREF and HNF of {{rket|{{map|33 0 -131}} {{map|0 33 131}}}}, DCF of {{rket|{{map|1 1 0}} {{map|0 33 131}}}}, and RREF of {{rket|{{map|1 0 <math>\frac{-131}{33}</math>}} {{map|0 1 <math>\frac{131}{33}</math>}}}}.
 That accounts for 7 of the 15 total possible cases for a system of equalities between 4 entities. The remaining 9 cases are impossible due to properties of the domain: