Normal forms
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An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matricies is [[http://Hermite normal form|Hermite normal form]], and by using Hermite normal form we may define normalized forms of lists of [[Harmonic Limit|p-limit]] musical intervals (or [[Monzos and Interval Space|monzos]]) or lists of [[Vals and Tuning Space|vals]], the interval (or monzo) normal list and the val normal list. There are slightly different definitions of Hermite normal form in use, and if you are using a computer proram to compute it, you should take care that the same monzo or val normal 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 firstly there can be defined 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 the nth row. H[F(i), i] will therefore be this first nonzero entry, and we have (1) If i < j, H[i, j] = 0 (H is upper triangular) (2) If F(i) > 0 then H[F(i), i] > 0; that is, the first nonzero entry in the ith column, counting up from the bottom, is positive. (3) If k > F(i) > 0 then H[k, i] = 0; that is, F(i) is the row of the first nonzero entry in the ith column, counting up from the bottom. (4) If F(i) > 0 then H[F(i), i] > H[F(i), k] >= 0 when i < k; that is, the first nonzero entry in the ith column, counting up from the bottom is greater than any of the rest along that row, which however are all non-negative. There is some redundency in the statement of these conditions, but that does no harm.
Original HTML content:
<html><head><title>Normal lists</title></head><body>An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matricies is [[<!-- ws:start:WikiTextUrlRule:18:http://Hermite --><a class="wiki_link_ext" href="http://Hermite" rel="nofollow">http://Hermite</a><!-- ws:end:WikiTextUrlRule:18 --> normal form|Hermite normal form]], and by using Hermite normal form we may define normalized forms of lists of <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> musical intervals (or <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a>) or lists of <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">vals</a>, the interval (or monzo) normal list and the val normal list.<br /> <br /> There are slightly different definitions of Hermite normal form in use, and if you are using a computer proram to compute it, you should take care that the same monzo or val normal list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows. <br /> <br /> An n by m integral matrix H is in Hermite normal form if firstly there can be defined 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 the nth row.<br /> H[F(i), i] will therefore be this first nonzero entry, and we have<br /> <br /> (1) If i < j, H[i, j] = 0 (H is upper triangular)<br /> <br /> (2) If F(i) > 0 then H[F(i), i] > 0; that is, the first nonzero entry in the ith column, counting up from the bottom, is positive.<br /> <br /> (3) If k > F(i) > 0 then H[k, i] = 0; that is, F(i) is the row of the first nonzero entry in the ith column, counting up from the bottom.<br /> <br /> (4) If F(i) > 0 then H[F(i), i] > H[F(i), k] >= 0 when i < k; that is, the first nonzero entry in the ith column, counting up from the bottom is greater than any of the rest along that row, which however are all non-negative.<br /> <br /> There is some redundency in the statement of these conditions, but that does no harm.</body></html>