Normal forms: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-15 01:58:58 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-15 02:07:39 UTC</tt>.

: The original revision id was <tt>~~142172347~~</tt>.

: The original revision id was <tt>142172695</tt>.

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There are slightly different definitions of Hermite normal form in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows.

An n by m integral matrix H is in Hermite normal form if when we define a function F such that F(i) = 0 if all of the entries in the ith column of H are 0, and otherwise F(i) is equal to the row number of the first nonzero entry in the ith column, ~~counting~~ up from the bottom, ie from the nth row, we have

An n by m integral matrix H is in Hermite normal form if when we define a function F such that F(i) = 0 if all of the entries in the ith column of H are 0, and otherwise F(i) is equal to the row number of the first nonzero entry in the ith column, checking up from the bottom, ie from the nth row, we have

(1) If i > j, H[i, j] = 0 (H is upper triangular.)

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There are slightly different definitions of Hermite normal form in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows.

An n by m integral matrix H is in Hermite normal form if when we define a function F such that F(i) = 0 if all of the entries in the ith column of H are 0, and otherwise F(i) is equal to the row number of the first nonzero entry in the ith column, ~~counting~~ up from the bottom, ie from the nth row, we have

An n by m integral matrix H is in Hermite normal form if when we define a function F such that F(i) = 0 if all of the entries in the ith column of H are 0, and otherwise F(i) is equal to the row number of the first nonzero entry in the ith column, checking up from the bottom, ie from the nth row, we have

(1) If i &gt; j, H[i, j] = 0 (H is upper triangular.)

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-15 01:58:58 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-15 02:07:39 UTC</tt>.<br>
-: The original revision id was <tt>142172347</tt>.<br>
+: The original revision id was <tt>142172695</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 10: / Line 10: @@
 There are slightly different definitions of Hermite normal form in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows.
-An n by m integral matrix H is in Hermite normal form if when we define a function F such that F(i) = 0 if all of the entries in the ith column of H are 0, and otherwise F(i) is equal to the row number of the first nonzero entry in the ith column, counting up from the bottom, ie from the nth row, we have
+An n by m integral matrix H is in Hermite normal form if when we define a function F such that F(i) = 0 if all of the entries in the ith column of H are 0, and otherwise F(i) is equal to the row number of the first nonzero entry in the ith column, checking up from the bottom, ie from the nth row, we have
 (1) If i &gt; j, H[i, j] = 0 (H is upper triangular.)
@@ Line 73: / Line 73: @@
 There are slightly different definitions of Hermite normal form in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows. &lt;br /&gt;
 &lt;br /&gt;
-An n by m integral matrix H is in Hermite normal form if when we define a function F such that F(i) = 0 if all of the entries in the ith column of H are 0, and otherwise F(i) is equal to the row number of the first nonzero entry in the ith column, counting up from the bottom, ie from the nth row, we have&lt;br /&gt;
+An n by m integral matrix H is in Hermite normal form if when we define a function F such that F(i) = 0 if all of the entries in the ith column of H are 0, and otherwise F(i) is equal to the row number of the first nonzero entry in the ith column, checking up from the bottom, ie from the nth row, we have&lt;br /&gt;
 &lt;br /&gt;
 (1) If i &amp;gt; j, H[i, j] = 0 (H is upper triangular.)&lt;br /&gt;