Dual list
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author genewardsmith and made on 2014-12-22 23:02:13 UTC.
- The original revision id was 535836140.
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Original Wikitext content:
If P is a matrix with rational coefficents, denote by Saturate(P) the integral matrix which is the [[saturation]] of P, defined in terms of the right reducing matrix to Smith normal form. We may clean the result up a little by LLL reduction, so let Sat(P) = LLL(Saturate(P)). Given an integral matrix A which we will view as a list of lists, A`A, where A` is the pseudoinverse of A, is the corresponding nxn projection matrix, where n. Also define Clear(A) to be the rational number matrix A cleared of denominators by multiplying through the rows by the LCM of the denominators, and Hrm(A) to be the Hermite reduction of A, with any rows consisting of zeros removed. We may combine these into the dual list function: Dulist(A) = Sat(Hrm(Clear(I - A`A))), where I is the rxr identity matrix, where r is the rank of A.
Original HTML content:
<html><head><title>dual list</title></head><body>If P is a matrix with rational coefficents, denote by Saturate(P) the integral matrix which is the <a class="wiki_link" href="/saturation">saturation</a> of P, defined in terms of the right reducing matrix to Smith normal form. We may clean the result up a little by LLL reduction, so let Sat(P) = LLL(Saturate(P)). Given an integral matrix A which we will view as a list of lists, A`A, where A` is the pseudoinverse of A, is the corresponding nxn projection matrix, where n. Also define Clear(A) to be the rational number matrix A cleared of denominators by multiplying through the rows by the LCM of the denominators, and Hrm(A) to be the Hermite reduction of A, with any rows consisting of zeros removed. We may combine these into the dual list function: Dulist(A) = Sat(Hrm(Clear(I - A`A))), where I is the rxr identity matrix, where r is the rank of A.</body></html>