Matrix echelon forms: Difference between revisions

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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>AKA 165b⁴c¹⁹&231b⁶c²⁴, which ~~tempers out~~ the 7.753¢ comma {{vector|-131 131 -33}}!</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:

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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>AKA 165b⁴c¹⁹&231b⁶c²⁴, which tempers out the 7.753¢ comma {{vector|-131 131 -33}}!</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: