#############################################################################
##
#W ffmat.xml
#Y Copyright (C) 2014-2022
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
<#GAPDoc Label="RowSpaceBasis">
<ManSection>
<Attr Name="RowSpaceBasis" Arg="m" Label="for a matrix over finite field"/>
<Attr Name="RowSpaceTransformation" Arg="m"
Label="for a matrix over finite field"/>
<Attr Name="RowSpaceTransformationInv" Arg="m"
Label="for a matrix over finite field"/>
<Description>
If <A>m</A> is a matrix object over a finite field, then
to compute the value of any of the above attributes, a canonical basis for
the row space of <A>m</A> is computed along with an invertible matrix
<C>RowSpaceTransformation</C> such that
<C>m * RowSpaceTransformation(m) = RowSpaceBasis(m)</C>.
<C>RowSpaceTransformationInv(m)</C> is the inverse
of <C>RowSpaceTransformation(m)</C>.
<Example><![CDATA[
gap> x := Matrix(GF(4), Z(4) ^ 0 * [[1, 1, 0], [0, 1, 1], [1, 1, 1]]);
[ [ Z(2)^0, Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0, Z(2)^0 ],
[ Z(2)^0, Z(2)^0, Z(2)^0 ] ]
gap> RowSpaceBasis(x);
<rowbasis of rank 3 over GF(2^2)>
gap> RowSpaceTransformation(x);
[ [ 0*Z(2), Z(2)^0, Z(2)^0 ], [ Z(2)^0, Z(2)^0, Z(2)^0 ],
[ Z(2)^0, 0*Z(2), Z(2)^0 ] ]]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="RightInverse">
<ManSection>
<Attr Name="RightInverse" Arg="m" Label="for a matrix over finite field"/>
<Attr Name="LeftInverse" Arg="m" Label="for a matrix over finite field"/>
<Returns>A matrix over a finite field.</Returns>
<Description>
These attributes contain a semigroup left-inverse, and a semigroup
right-inverse of the matrix <A>m</A> respectively.
<Example><![CDATA[
gap> x := Matrix(GF(4), Z(4) ^ 0 * [[1, 1, 0], [0, 0, 0], [1, 1, 1]]);
[ [ Z(2)^0, Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2) ],
[ Z(2)^0, Z(2)^0, Z(2)^0 ] ]
gap> LeftInverse(x);
[ [ Z(2)^0, Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2) ],
[ Z(2)^0, 0*Z(2), Z(2)^0 ] ]
gap> Display(LeftInverse(x) * x);
1 1 .
. . .
. . 1
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
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