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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains methods for *matrix modules*, that is,
## free left modules consisting of matrices.
##
## Especially methods for *full matrix modules* $R^[m,n]$ are contained.
##
## (See the file `modulrow.gi' for the methods for row modules.
## Note that we do not need methods for enumerator and iterator of full
## matrix modules because the standard delegation to full row modules
## suffices.)
##
#############################################################################
##
#F FullMatrixModule( <R>, <m>, <n> )
##
## Let $E_{i,j}$ be the matrix with value 1 in row $i$ and $column $j$, and
## zero otherwise.
## Clearly the full matrix space is generated by all $E_{i,j}$, $i$ and
## $j$ ranging from 1 to <m> and <n>, respectively.
##
## `FullMatrixModule' returns a module of ordinary matrices
## (not of Lie matrices, see~"IsOrdinaryMatrix").
##
InstallGlobalFunction( FullMatrixModule, function( R, m, n )
local M; # the free module record, result
if not ( IsRing( R ) and IsInt( m ) and 0 <= m
and IsInt( n ) and 0 <= n ) then
Error( "usage: FullMatrixModule( <R>, <m>, <n> ) for ring <R>" );
elif m = n then
return FullMatrixFLMLOR( R, m );
fi;
if IsDivisionRing( R ) then
M:= Objectify( NewType( CollectionsFamily( CollectionsFamily(
FamilyObj( R ) ) ),
IsFreeLeftModule
and IsGaussianSpace
and IsFullMatrixModule
and IsAttributeStoringRep ),
rec() );
else
M:= Objectify( NewType( CollectionsFamily( CollectionsFamily(
FamilyObj( R ) ) ),
IsFreeLeftModule
and IsFullMatrixModule
and IsAttributeStoringRep ),
rec() );
fi;
SetLeftActingDomain( M, R );
SetDimensionOfVectors( M, [ m, n ] );
return M;
end );
#############################################################################
##
#M \^( <M>, [ <m>, <n> ] ) . . . . . . . . . full matrix module over a ring
##
InstallOtherMethod( \^,
"for ring and list of integers (delegate to `FullMatrixModule')",
[ IsRing, IsCyclotomicCollection and IsList ],
function( R, n )
if Length( n ) = 2
and IsInt( n[1] ) and 0 <= n[1]
and IsInt( n[2] ) and 0 <= n[2] then
return FullMatrixModule( R, n[1], n[2] );
fi;
TryNextMethod();
end );
#############################################################################
##
#M IsMatrixModule( <M> )
##
InstallMethod( IsMatrixModule,
"for a free left module",
[ IsFreeLeftModule and HasGeneratorsOfLeftModule ],
function( M )
local gens;
gens:= GeneratorsOfLeftModule( M );
if IsEmpty( gens ) then
return IsMatrix( Zero( M ) );
else
return ForAll( gens, IsMatrix );
fi;
end );
InstallMethod( IsMatrixModule,
"for a free left module without generators",
[ IsFreeLeftModule ],
M -> IsMatrix(Representative(M)));
#############################################################################
##
#M IsMatrixModule . return `false' for objects which are not free left modules
##
InstallOtherMethod( IsMatrixModule,
"return `false' for objects which are not free left modules",
true, [ IsObject ], 0,
function ( obj )
if not IsFreeLeftModule(obj) then
return false;
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M IsFullMatrixModule( M )
##
InstallMethod( IsFullMatrixModule,
"for matrix module",
[ IsFreeLeftModule ],
M -> IsMatrixModule( M )
and Dimension( M ) = Product( DimensionOfVectors( M ) )
and ForAll( GeneratorsOfLeftModule( M ),
v -> IsSubset( LeftActingDomain( M ), v ) ) );
#############################################################################
##
#M Dimension( <M> )
##
InstallMethod( Dimension,
"for full matrix module",
[ IsFreeLeftModule and IsFullMatrixModule ],
M -> Product( DimensionOfVectors( M ) ) );
#############################################################################
##
#M Random( <M> )
##
InstallMethodWithRandomSource( Random,
"for a random source and a full matrix module",
[ IsRandomSource, IsFreeLeftModule and IsFullMatrixModule ],
function( rs, M )
local random;
random:= DimensionOfVectors( M );
random:= RandomMat( rs, random[1], random[2],
LeftActingDomain( M ) );
if IsLieObjectCollection( M ) then
random:= LieObject( random );
fi;
return random;
end );
#############################################################################
##
#M Representative( <M> )
##
InstallMethod( Representative,
"for full matrix module",
[ IsFreeLeftModule and IsFullMatrixModule ],
function( M )
local random;
random:= DimensionOfVectors( M );
return NullMat( random[1], random[2], LeftActingDomain( M ) );
end );
#############################################################################
##
#F StandardGeneratorsOfFullMatrixModule( <M> )
##
InstallGlobalFunction( StandardGeneratorsOfFullMatrixModule, function( M )
local R, one, dims, m, n, zeromat, gens, i, j, gen;
R:= LeftActingDomain( M );
one:= One( R );
dims:= DimensionOfVectors( M );
m:= dims[1];
n:= dims[2];
zeromat:= NullMat( m, n, R );
gens:= [];
for i in [ 1 .. m ] do
for j in [ 1 .. n ] do
gen:= List( zeromat, ShallowCopy );
gen[i,j]:= one;
Add( gens, gen );
od;
od;
if IsLieObjectCollection( M ) then
gens:= List( gens, LieObject );
fi;
return gens;
end );
#############################################################################
##
#M GeneratorsOfLeftModule( <V> )
##
InstallMethod( GeneratorsOfLeftModule,
"for full matrix module",
[ IsFreeLeftModule and IsFullMatrixModule ],
StandardGeneratorsOfFullMatrixModule );
#############################################################################
##
#M ViewObj( <M> )
##
InstallMethod( ViewObj,
"for full matrix module",
[ IsFreeLeftModule and IsFullMatrixModule ],
function( M )
if IsLieObjectCollection( M ) then
TryNextMethod();
fi;
Print( "( " );
View( LeftActingDomain( M ) );
Print( "^", DimensionOfVectors( M ), " )" );
end );
#############################################################################
##
#M PrintObj( <M> )
##
InstallMethod( PrintObj,
"for full matrix module",
[ IsFreeLeftModule and IsFullMatrixModule ],
function( M )
if IsLieObjectCollection( M ) then
TryNextMethod();
fi;
Print( "( ", LeftActingDomain( M ), "^", DimensionOfVectors( M ), " )" );
end );
#############################################################################
##
#M \in( <v>, <V> )
##
InstallMethod( \in,
"for full matrix module",
IsElmsColls,
[ IsObject,
IsFreeLeftModule and IsFullMatrixModule ],
function( mat, M )
return IsMatrix( mat )
and IsRectangularTable( mat )
and DimensionsMat( mat ) = DimensionOfVectors( M )
and ForAll( mat, row -> IsSubset( LeftActingDomain( M ), row ) );
end );
#############################################################################
##
#M BasisVectors( <B> ) . . . . for a canonical basis of a full matrix module
##
InstallMethod( BasisVectors,
"for canonical basis of a full matrix module",
[ IsBasis and IsCanonicalBasis and IsCanonicalBasisFullMatrixModule ],
B -> StandardGeneratorsOfFullMatrixModule( UnderlyingLeftModule( B ) ) );
#############################################################################
##
#M CanonicalBasis( <V> )
##
InstallMethod( CanonicalBasis,
[ IsFreeLeftModule and IsFullMatrixModule ],
function( V )
local B;
B:= Objectify( NewType( FamilyObj( V ),
IsFiniteBasisDefault
and IsCanonicalBasis
and IsCanonicalBasisFullMatrixModule
and IsAttributeStoringRep ),
rec() );
SetUnderlyingLeftModule( B, V );
return B;
end );
#############################################################################
##
#M Basis( <M> ) . . . . . . . . . . . . . . . . . . for full matrix module
##
InstallMethod( Basis,
"for full matrix module (delegate to `CanonicalBasis')",
[ IsFreeLeftModule and IsFullMatrixModule ], CANONICAL_BASIS_FLAGS,
CanonicalBasis );
#############################################################################
##
#M Coefficients( <B>, <m> ) . for a canonical basis of a full matrix module
##
InstallMethod( Coefficients,
"for canonical basis of a full matrix module",
IsCollsElms,
[ IsBasis and IsCanonicalBasisFullMatrixModule, IsMatrix ],
function( B, mat )
local V, R;
V:= UnderlyingLeftModule( B );
R:= LeftActingDomain( V );
if DimensionsMat( mat ) = DimensionOfVectors( V )
and ForAll( mat, row -> IsSubset( R, row ) ) then
return Concatenation( mat );
else
return fail;
fi;
end );
#############################################################################
##
#M IsCanonicalBasisFullMatrixModule( <B> ) . . . . . . . . . . . for a basis
##
InstallMethod( IsCanonicalBasisFullMatrixModule,
"for a basis",
[ IsBasis ],
B -> IsFullMatrixModule( UnderlyingLeftModule( B ) )
and IsCanonicalBasis( B ) );
[ Dauer der Verarbeitung: 0.31 Sekunden
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