1zqt(1)                           User Commands                          zqt(1)
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NAME

6       zqt - clean and quotient on a matrix
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SYNOPSIS

9       zqt [OPTIONS] <Subsp> <Matrix> <Quotient>
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DESCRIPTION

12       This  program  reads in a subspace and applies the canonical map to its
13       quotient on a matrix.  The result is written out to Quot.  Subsp should
14       be  a matrix in semi-echelon form, and the two input matrices must have
15       the same field parameter and the same number of columns.   If  this  is
16       not the case the program stops with an error message.
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18       Otherwise  the  program reads in Subsp, builds a table of pivot columns
19       and then proceeds, row by row, through Matrix.  For each row, the  sig‐
20       nificant entries are zeroized by adding the correct multiple of rows of
21       Subsp.  The insignificant columns are then extracted and written out to
22       Quot.  Hence
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24       · Subsp has M rows, N columns and is in echelon form,
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26       · Matrix has L rows, N columns and is otherwise arbitrary, and
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28       · Quot has L rows and N-M columns.
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30       In  other  words,  the program calculates the projection of Matrix onto
31       the B-A dimensional quotient space defined by Subsp.  If the -i  option
32       is  used, zqt calculates the action of Matrix on the quotient.  This is
33       done by projecting the matrix as explained above, and taking  only  the
34       insignificant  rows.   Insignificant  rows  are defined by treating the
35       pivot table as a table of rows rather than columns.  Example: Let "spc"
36       be an invariant subspace and "z1" an algebra element (a square matrix).
37       Then, after
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39         zqt -i spc z1 q1</pre>
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41       "q1" contains the action of "z1" on the quotient by "spc".
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43       Another, less obvious use of zqt is to condense  a  matrix  representa‐
44       tion.   First, find an element E of the group algebra with stable rank,
45       i.e., rank(E*E) = rank(E).  This can be done by taking any element F of
46       the group algebra and raising it to higher powers until the rank stabi‐
47       lizes.  We may then condense onto the kernel of E as follows:
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49       zef E X        X is the echelon form of Image(E)
50       znu E Y        Y is the kernel of E
51       zqt X Y Z      calculate the canonical projection of Y ...
52       ziv Z T        ... and adjust Y so that the canonical ...
53       zmu T Y Y1     ... projection of Y1 is the identity
54       zmu Y1 Z1 T1   calculate KZ1 = condensed Z1
55       zqt X T1 KZ1
56       zmu Y1 Z2 T1   calculate KZ2 = condensed Z2
57       zqt X T1 KZ2
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OPTIONS

60       -Q     Quiet, no messages.
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62       -V     Verbose, more messages.
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64       -T <MaxTime>
65              Set CPU time limit
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67       -i     Take only insignificant rows of Matrix.  Quotient  will  be  the
68              action of Matrix on the quotient by subspace Subsp.
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IMPLEMENTATION DETAILS

71       It is not completely checked that Subsp is in echelon form.
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73       The Subspace and one row of both Matrix and Subsp must fit into memory.
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SEE ALSO

76       zef(1), ziv(1), zmu(1), znu(1)
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80MeatAxe                             2.4.24                              zqt(1)