Defactoring: Difference between revisions

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# all entries in pivot columns above the pivots ≥ 0 and strictly less than the pivot

One special thing about computing the HNF is that we're not allowed to use all elementary operations; in particular we're not allowed to multiply (or divide) rows. Our main technique, then, will be adding or subtract rows from each other. This, of course, includes adding or subtracting ''multiples'' of rows from each other, because doing so is equivalent to performing those additions or subtractions one at a time (note that adding or subtracting ''multiples'' of rows from each other is significantly different than simply ''multiplying'' a row by itself).

One special thing about computing the HNF is that we're not allowed to use all elementary operations; in particular we're not allowed to multiply (or divide) rows. Our main technique, then, will be adding or subtract rows from each other. This, of course, includes adding or subtracting ''multiples'' of rows from each other, because doing so is equivalent to performing those additions or subtractions one at a time (note that adding or subtracting ''multiples'' of rows from each other is significantly different than simply ''multiplying'' a row by itself).<ref>The fact that you're not allowed to multiply or divide is equivalent to the fact that at every step along the way, the augmented matrix remains unimodular.</ref>

So let's begin by subtracting the 1st row from the 2nd row, and let's do it 2 times, because we can see that would get the pivot of the 2nd row pretty close to 0, which is where we're trying to get it, per the 2nd HNF constraint above.

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 # all entries in pivot columns above the pivots ≥ 0 and strictly less than the pivot
-One special thing about computing the HNF is that we're not allowed to use all elementary operations; in particular we're not allowed to multiply (or divide) rows. Our main technique, then, will be adding or subtract rows from each other. This, of course, includes adding or subtracting ''multiples'' of rows from each other, because doing so is equivalent to performing those additions or subtractions one at a time (note that adding or subtracting ''multiples'' of rows from each other is significantly different than simply ''multiplying'' a row by itself).
+One special thing about computing the HNF is that we're not allowed to use all elementary operations; in particular we're not allowed to multiply (or divide) rows. Our main technique, then, will be adding or subtract rows from each other. This, of course, includes adding or subtracting ''multiples'' of rows from each other, because doing so is equivalent to performing those additions or subtractions one at a time (note that adding or subtracting ''multiples'' of rows from each other is significantly different than simply ''multiplying'' a row by itself).<ref>The fact that you're not allowed to multiply or divide is equivalent to the fact that at every step along the way, the augmented matrix remains unimodular.</ref>
 So let's begin by subtracting the 1st row from the 2nd row, and let's do it 2 times, because we can see that would get the pivot of the 2nd row pretty close to 0, which is where we're trying to get it, per the 2nd HNF constraint above.