Defactoring: Difference between revisions

Line 1,029:

So HNF has a lot in common with IRREF, which is the IREF you find by converting the RREF, but it is not always the same as IRREF.

~~=== DCF ===~~

Canonical form is closely related to HNF, because the second step of finding the DCF is taking the HNF. So the DCF is always ''a'' HNF, and therefore it has all the same properties of being echelon, integer, and normalized, and in turn therefore it also provides a unique representation. However it is not necessary ''the'' same HNF of the original mapping, due to the first step being defactoring; it is the same as as HNF except when the original mapping is enfactored.

~~'''Defactored~~ Canonical ~~Form''', or '''DCF''' (listed here not because it predates itself of course, but for completeness)~~ is closely related to HNF, because the second step of finding the DCF is taking the HNF. So the DCF is always ''a'' HNF, and therefore it has all the same properties of being echelon, integer, and normalized, and in turn therefore it also provides a unique representation. However it is not necessary ''the'' same HNF of the original mapping, due to the first step being defactoring; it is the same as as HNF except when the original mapping is enfactored.

In the below example, <math>p_{ij}</math> represents any positive integer, and <math>a_{ij}</math> represents any nonnegative integer less than the <math>p</math> in its column.

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

|+HNF and DFC

|+HNF (and DFC)

|style="width: 25px; background-color: #e69138;"|p₁₁

|style="width: 25px; background-color: #ffe599;"|a₁₂

@@ Line 1,029: / Line 1,029: @@
 So HNF has a lot in common with IRREF, which is the IREF you find by converting the RREF, but it is not always the same as IRREF.
-=== DCF ===
+Canonical form is closely related to HNF, because the second step of finding the DCF is taking the HNF. So the DCF is always ''a'' HNF, and therefore it has all the same properties of being echelon, integer, and normalized, and in turn therefore it also provides a unique representation. However it is not necessary ''the'' same HNF of the original mapping, due to the first step being defactoring; it is the same as as HNF except when the original mapping is enfactored.
-'''Defactored Canonical Form''', or '''DCF''' (listed here not because it predates itself of course, but for completeness) is closely related to HNF, because the second step of finding the DCF is taking the HNF. So the DCF is always ''a'' HNF, and therefore it has all the same properties of being echelon, integer, and normalized, and in turn therefore it also provides a unique representation. However it is not necessary ''the'' same HNF of the original mapping, due to the first step being defactoring; it is the same as as HNF except when the original mapping is enfactored.
 In the below example, <span><math>p_{ij}</math></span> represents any positive integer, and <span><math>a_{ij}</math></span> represents any nonnegative integer less than the <span><math>p</math></span> in its column.
 {| class="wikitable" style="text-align: center;"
-|+HNF and DFC
+|+HNF (and DFC)
 |style="width: 25px; background-color: #e69138;"|p₁₁
 |style="width: 25px; background-color: #ffe599;"|a₁₂