Hodge dual
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- This revision was by author genewardsmith and made on 2012-01-01 21:56:59 UTC.
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Original Wikitext content:
Given a k-multival W, there is a //dual// k-multimonzo Wº. Similarly, given a k-multimonzo M, there is a dual k-multival Mº. The dual may be defined in terms of the bracket product relating multivals and multimonzos, which we discuss first. Given a k-multival W and a k-multimonzo M (in which we may include sums of k-fold wedge products of vals or monzos), the bracket or bracket product, <W|M>, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product W = 612∧441 = <<18 27 18 1 -22 -34||, which is the wedgie for ennealimmal temperament, and is a 2-val. The suppose we take the wedge product of the monzos for 27/25 and 21/20, M = |0 3 -2 0>∧|-2 1 -1 1> = ||6 -4 0 -1 3 -2>>. Them <W|M> = <<18 27 18 1 -22 -34||6 -4 0 -1 3 -2>> = 18*6-27*4+18*0-1*1-22*3+34*2 = 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the [[interior product]], but then we must fuss about the sign. Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coeifficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that <U|Vº> = U∧V for every k-multival U.
Original HTML content:
<html><head><title>The dual</title></head><body>Given a k-multival W, there is a <em>dual</em> k-multimonzo Wº. Similarly, given a k-multimonzo M, there is a dual k-multival Mº. The dual may be defined in terms of the bracket product relating multivals and multimonzos, which we discuss first.<br /> <br /> Given a k-multival W and a k-multimonzo M (in which we may include sums of k-fold wedge products of vals or monzos), the bracket or bracket product, <W|M>, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product W = 612∧441 = <<18 27 18 1 -22 -34||, which is the wedgie for ennealimmal temperament, and is a 2-val. The suppose we take the wedge product of the monzos for 27/25 and 21/20, M = |0 3 -2 0>∧|-2 1 -1 1> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x||6 -4 0 -1 3 -2>>. Them"></a><!-- ws:end:WikiTextHeadingRule:0 --> ||6 -4 0 -1 3 -2>>. Them <W|M> </h1> <<18 27 18 1 -22 -34||6 -4 0 -1 3 -2>> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="x18*6-27*4+18*0-1*1-22*3+34*2"></a><!-- ws:end:WikiTextHeadingRule:2 --> 18*6-27*4+18*0-1*1-22*3+34*2 </h1> 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the <a class="wiki_link" href="/interior%20product">interior product</a>, but then we must fuss about the sign. <br /> <br /> Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coeifficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that <U|Vº> = U∧V for every k-multival U.</body></html>