Matrix echelon forms: Difference between revisions

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The mathematical field of [[~~linear algebra formalism|~~linear algebra]] has defined a number of echelon forms for matrices, many of them have been used in [[regular temperament theory]]. This article is intended as a reference and gathering place for relevant information about these matrix forms.

The mathematical field of [[linear algebra]] has defined a number of echelon forms for matrices, many of them have been used in [[regular temperament theory]]. This article is intended as a reference and gathering place for relevant information about these matrix forms.

== Row echelon form ==

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# 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 group="note">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 {{frac|−131|33}}}}}} {{map|0 1 {{frac|131|33}}}}}}.

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 group="note">AKA 165b4c19&231b6c24, which makes the 7.753{{cent}} 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 {{frac|−131|33}}}}}} {{map|0 1 {{frac|131|33}}}}}}.

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:

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== References ==

[[Category:Regular temperament theory]]

[[Category:Terms]]

[[Category:Math]]

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-The mathematical field of [[linear algebra formalism|linear algebra]] has defined a number of echelon forms for matrices, many of them have been used in [[regular temperament theory]]. This article is intended as a reference and gathering place for relevant information about these matrix forms.
+The mathematical field of [[linear algebra]] has defined a number of echelon forms for matrices, many of them have been used in [[regular temperament theory]]. This article is intended as a reference and gathering place for relevant information about these matrix forms.
 == Row echelon form ==
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 # 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 group="note">AKA 165b<sup>4</sup>c<sup>19</sup>&231b<sup>6</sup>c<sup>24</sup>, 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 {{frac|−131|33}}}}}} {{map|0 1 {{frac|131|33}}}}}}.
+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 group="note">AKA 165b<sup>4</sup>c<sup>19</sup>&231b<sup>6</sup>c<sup>24</sup>, which makes the 7.753{{cent}} 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 {{frac|−131|33}}}}}} {{map|0 1 {{frac|131|33}}}}}}.
 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:
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 == References ==
-<references/>
+<references />
 [[Category:Regular temperament theory]]
 [[Category:Terms]]
 [[Category:Math]]