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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346#############################################################################
##
## HomalgMatrix.gi MatricesForHomalg package Mohamed Barakat
##
## Copyright 2007-2008 Lehrstuhl B für Mathematik, RWTH Aachen
##
## Implementations for homalg matrices.
##
#############################################################################
####################################
#
# representations:
#
####################################
##
DeclareRepresentation( "IshomalgInternalMatrixHullRep",
IsInternalMatrixHull,
[ ] );
## <#GAPDoc Label="IsHomalgInternalMatrixRep">
## <ManSection>
## <Filt Type="Representation" Arg="A" Name="IsHomalgInternalMatrixRep"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## The internal representation of &homalg; matrices. <P/>
## (It is a representation of the &GAP; category <Ref Filt="IsHomalgMatrix"/>.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareRepresentation( "IsHomalgInternalMatrixRep",
IsHomalgMatrix,
[ ] );
####################################
#
# families and types:
#
####################################
# two new family:
BindGlobal( "TheFamilyOfInternalMatrixHulls",
NewFamily( "TheFamilyOfInternalMatrixHulls" ) );
BindGlobal( "TheFamilyOfHomalgMatrices",
NewFamily( "TheFamilyOfHomalgMatrices" ) );
# two new types:
BindGlobal( "TheTypeInternalMatrixHull",
NewType( TheFamilyOfInternalMatrixHulls,
IshomalgInternalMatrixHullRep ) );
BindGlobal( "TheTypeHomalgInternalMatrix",
NewType( TheFamilyOfHomalgMatrices,
IsHomalgInternalMatrixRep ) );
####################################
#
# compatibility code for the new
# IsMatrixObj interface
#
####################################
InstallMethod( Length,
[ IsHomalgMatrix ], 0,
NrColumns );
####################################
#
# methods for operations:
#
####################################
##-----------------------------------------------------------------------------
#
# put all methods to trace errors in LIMAT.gi with the very high priority 10001
#
##-----------------------------------------------------------------------------
##
InstallMethod( Rank,
"for homalg matrices",
[ IsInternalMatrixHull ],
function( M )
return Rank( M!.matrix );
end );
## <#GAPDoc Label="HomalgRing:matrix">
## <ManSection>
## <Oper Arg="mat" Name="HomalgRing" Label="for matrices"/>
## <Returns>a &homalg; ring</Returns>
## <Description>
## The &homalg; ring of the &homalg; matrix <A>mat</A>.
## <Example><![CDATA[
## gap> ZZ := HomalgRingOfIntegers( );
## Z
## gap> d := HomalgDiagonalMatrix( [ 2 .. 4 ], ZZ );
## <An unevaluated diagonal 3 x 3 matrix over an internal ring>
## gap> R := HomalgRing( d );
## Z
## gap> IsIdenticalObj( R, ZZ );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( HomalgRing,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return M!.ring;
end );
##
InstallMethod( BaseDomain,
"for homalg matrices",
[ IsHomalgMatrix ],
HomalgRing );
##
InstallMethod( BlindlyCopyMatrixProperties, ## under construction
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( S, T )
## if the new ring only interprets the 1x1 submatrices as elements
## then it is safe to at least copy the following attributes
if HasNrRows( S ) then
SetNrRows( T, NrRows( S ) );
fi;
if HasNrColumns( S ) then
SetNrColumns( T, NrColumns( S ) );
fi;
end );
##
InstallMethod( ShallowCopy,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, MM;
R := HomalgRing( M );
RP := homalgTable( R );
if IsBound(RP!.ShallowCopy) then
MM := HomalgMatrixWithAttributes( [
Eval, RP!.ShallowCopy( M ),
NrRows, NrRows( M ),
NrColumns, NrColumns( M ),
], R );
MatchPropertiesAndAttributes( M, MM,
LIMAT.intrinsic_properties,
LIMAT.intrinsic_attributes,
LIMAT.intrinsic_components
);
return MM;
fi;
## we have no other choice
return M;
end );
##
InstallMethod( MutableCopyMat,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, MM;
R := HomalgRing( M );
RP := homalgTable( R );
if IsBound(RP!.ShallowCopy) then
MM := HomalgMatrixWithAttributes( [
Eval, RP!.ShallowCopy( M ),
NrRows, NrRows( M ),
NrColumns, NrColumns( M ),
], R );
SetIsMutableMatrix( MM, true );
return MM;
fi;
## we have no other choice
TryNextMethod( );
end );
##
InstallMethod( ShallowCopy,
"for homalg internal matrices",
[ IsHomalgInternalMatrixRep ],
function( M )
local R, RP, MM;
R := HomalgRing( M );
RP := homalgTable( R );
if IsBound(RP!.ShallowCopy) then
MM := HomalgMatrixWithAttributes( [
Eval, RP!.ShallowCopy( M ),
NrRows, NrRows( M ),
NrColumns, NrColumns( M ),
], R );
if not IsIdenticalObj( Eval( M ), Eval( MM ) ) then
MatchPropertiesAndAttributes( M, MM,
LIMAT.intrinsic_properties,
LIMAT.intrinsic_attributes,
LIMAT.intrinsic_components
);
return MM;
fi;
fi;
if not IsInternalMatrixHull( Eval( M ) ) then
TryNextMethod( );
fi;
return HomalgMatrix( One( R ) * Eval( M )!.matrix, NrRows( M ), NrColumns( M ), R );
end );
##
InstallMethod( MutableCopyMat,
"for homalg internal matrices",
[ IsHomalgInternalMatrixRep ],
function( M )
local R, RP, MM;
R := HomalgRing( M );
RP := homalgTable( R );
if IsBound(RP!.ShallowCopy) then
MM := HomalgMatrixWithAttributes( [
Eval, RP!.ShallowCopy( M ),
NrRows, NrRows( M ),
NrColumns, NrColumns( M ),
], R );
if not IsIdenticalObj( Eval( M ), Eval( MM ) ) then
SetIsMutableMatrix( MM, true );
return MM;
fi;
fi;
if not IsInternalMatrixHull( Eval( M ) ) then
TryNextMethod( );
fi;
MM := HomalgMatrix( One( R ) * Eval( M )!.matrix, NrRows( M ), NrColumns( M ), R );
SetIsMutableMatrix( MM, true );
return MM;
end );
##
InstallMethod( ShallowCopy,
"for homalg matrices",
[ IsHomalgMatrix and IsOne ],
function( M )
return HomalgIdentityMatrix( NrRows( M ), HomalgRing( M ) );
end );
##
InstallMethod( MutableCopyMat,
"for homalg matrices",
[ IsHomalgMatrix and IsOne ],
function( M )
## do not use HomalgIdentityMatrix since
## we might want to alter the result
return HomalgInitialIdentityMatrix( NrRows( M ), HomalgRing( M ) );
end );
##
InstallMethod( ShallowCopy,
"for homalg matrices",
[ IsHomalgMatrix and IsZero ],
function( M )
return HomalgZeroMatrix( NrRows( M ), NrColumns( M ), HomalgRing( M ) );
end );
##
InstallMethod( MutableCopyMat,
"for homalg matrices",
[ IsHomalgMatrix and IsZero ],
function( M )
## do not use HomalgZeroMatrix since
## we might want to alter the result
return HomalgInitialMatrix( NrRows( M ), NrColumns( M ), HomalgRing( M ) );
end );
##
InstallMethod( SetConvertHomalgMatrixViaSparseString,
"for homalg matrices",
[ IsHomalgMatrix, IsBool ],
function( M, b )
M!.ConvertHomalgMatrixViaSparseString := b;
end );
##
InstallMethod( SetConvertHomalgMatrixViaFile,
"for homalg matrices",
[ IsHomalgMatrix, IsBool ],
function( M, b )
M!.ConvertHomalgMatrixViaFile := b;
end );
##
InstallMethod( SetMatElm,
"for homalg matrices",
[ IsHomalgMatrix and IsMutable, IsPosInt, IsPosInt, IsString, IsHomalgRing ],
function( M, r, c, s, R )
SetMatElm( M, r, c, s / R, R );
end );
##
InstallMethod( SetMatElm,
"for homalg internal matrices",
[ IsHomalgInternalMatrixRep and IsMutable, IsPosInt, IsPosInt, IsString, IsHomalgInternalRingRep ],
function( M, r, c, s, R )
SetMatElm( M, r, c, One( R ) * EvalString( s ), R );
end );
##
InstallMethod( SetMatElm,
"for homalg matrices",
[ IsHomalgMatrix, IsPosInt, IsPosInt, IsString ],
function( M, r, c, s )
Error( "the homalg matrix is write-protected\n" );
end );
##
InstallMethod( SetMatElm,
"for homalg matrices",
[ IsHomalgMatrix and IsMutable, IsPosInt, IsPosInt, IsString ],
function( M, r, c, s )
SetMatElm( M, r, c, s, HomalgRing( M ) );
end );
##
InstallMethod( SetMatElm,
"for homalg internal matrices",
[ IsHomalgInternalMatrixRep and IsMutable, IsPosInt, IsPosInt, IsRingElement, IsHomalgInternalRingRep ],
function( M, r, c, a, R )
if not IsInternalMatrixHull( Eval( M ) ) then
TryNextMethod( );
fi;
Eval( M )!.matrix[r][c] := One( R ) * a;
end );
##
InstallMethod( SetMatElm,
"for homalg matrices",
[ IsHomalgMatrix, IsPosInt, IsPosInt, IsRingElement ],
function( M, r, c, a )
Error( "the homalg matrix is write-protected\n" );
end );
##
InstallMethod( SetMatElm,
"for homalg matrices",
[ IsHomalgMatrix and IsMutable, IsPosInt, IsPosInt, IsRingElement ],
function( M, r, c, a )
SetMatElm( M, r, c, a, HomalgRing( M ) );
end );
##
InstallMethod( AddToMatElm,
"for homalg matrices",
[ IsHomalgMatrix, IsPosInt, IsPosInt, IsRingElement ],
function( M, r, c, a )
Error( "the homalg matrix is write-protected\n" );
end );
##
InstallMethod( AddToMatElm,
"for homalg matrices",
[ IsHomalgMatrix and IsMutable, IsPosInt, IsPosInt, IsRingElement, IsHomalgRing ],
function( M, r, c, a, R )
SetMatElm( M, r, c, a + MatElm( M, r, c, R ), R );
end );
##
InstallMethod( AddToMatElm,
"for homalg matrices",
[ IsHomalgMatrix and IsMutable, IsPosInt, IsPosInt, IsRingElement ],
function( M, r, c, a )
AddToMatElm( M, r, c, a, HomalgRing( M ) );
end );
##
InstallMethod( MatElmAsString,
"for homalg internal matrices",
[ IsHomalgInternalMatrixRep, IsPosInt, IsPosInt, IsHomalgInternalRingRep ],
function( M, r, c, R )
return String( MatElm( M, r, c ) );
end );
##
InstallMethod( MatElmAsString,
"for homalg matrices",
[ IsHomalgMatrix, IsPosInt, IsPosInt ],
function( M, r, c )
return MatElmAsString( M, r, c, HomalgRing( M ) );
end );
##
InstallMethod( MatElm,
"for homalg internal matrices",
[ IsHomalgInternalMatrixRep, IsPosInt, IsPosInt, IsHomalgInternalRingRep ],
function( M, r, c, R )
if IsInternalMatrixHull( Eval( M ) ) then
return Eval( M )!.matrix[r][c];
fi;
TryNextMethod( );
end );
##
InstallMethod( MatElm,
"for homalg matrices",
[ IsHomalgMatrix, IsPosInt, IsPosInt ],
function( M, r, c )
return MatElm( M, r, c, HomalgRing( M ) );
end );
##
InstallMethod( GetListOfMatrixAsString,
"for matrices",
[ IsList ],
function( M )
M := List( M, row -> List( row, String ) );
M := List( M, JoinStringsWithSeparator );
M := JoinStringsWithSeparator( M );
return Concatenation( "[", M, "]" );
end );
##
InstallMethod( GetListListOfStringsOfMatrix,
"for matrices and a homalg internal ring",
[ IsList, IsHomalgInternalRingRep ],
function( M, R )
local c, d, z;
if not ForAll( M, IsList ) then
TryNextMethod( );
fi;
if not HasCharacteristic( R ) then
Error( "characteristic not set\n" );
fi;
c := Characteristic( R );
if HasCoefficientsRing( R ) then
TryNextMethod( );
elif c = 0 then
return List( M, a -> List( a, String ) );
elif IsPrime( c ) then
if HasDegreeOverPrimeField( R ) and DegreeOverPrimeField( R ) > 1 then
d := DegreeOverPrimeField( R );
z := R!.NameOfPrimitiveElement;
return List( M, a -> List( a, b -> FFEToString( b, c, d, z ) ) );
fi;
return List( M, a -> List( a, b -> String( IntFFE( b ) ) ) );
fi;
return List( M, a -> List( a, b -> String( b![1] ) ) );
end );
##
InstallMethod( GetListListOfStringsOfHomalgMatrix,
"for homalg internal matrices",
[ IsHomalgInternalMatrixRep, IsHomalgInternalRingRep ],
function( M, R )
if not IsInternalMatrixHull( Eval( M ) ) then
TryNextMethod( );
fi;
return GetListListOfStringsOfMatrix( Eval( M )!.matrix, R );
end );
##
InstallMethod( GetListOfHomalgMatrixAsString,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return GetListOfHomalgMatrixAsString( M, HomalgRing( M ) );
end );
##
InstallMethod( GetListOfHomalgMatrixAsString,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgRing ],
function( M, R )
local c, s;
c := NrColumns( M );
s := List( [ 1 .. NrRows( M ) ], i -> List( [ 1 .. c ], j -> MatElmAsString( M, i, j ) ) );
s := JoinStringsWithSeparator( Concatenation( s ) );
return Concatenation( "[", s, "]" );
end );
##
InstallMethod( GetListOfHomalgMatrixAsString,
"for homalg internal matrices",
[ IsHomalgInternalMatrixRep, IsHomalgInternalRingRep ],
function( M, R )
local s, m;
if not IsInternalMatrixHull( Eval( M ) ) then
TryNextMethod( );
fi;
s := Eval( M )!.matrix;
if HasCharacteristic( R ) then
m := Characteristic( R );
if m > 0 and not HasCoefficientsRing( R ) then ## FIXME: we can only deal with Z/mZ and GF(p): m = Size( R ) !!!
if IsPrime( m ) then
s := List( s, a -> List( a, IntFFE ) );
else
s := List( s, a -> List( a, b -> b![1] ) );
fi;
fi;
fi;
s := String( Concatenation( s ) );
RemoveCharacters( s, "\\\n " );
return s;
end );
##
InstallMethod( GetListListOfHomalgMatrixAsString,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return GetListListOfHomalgMatrixAsString( M, HomalgRing( M ) );
end );
##
InstallMethod( GetListListOfHomalgMatrixAsString,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgRing ],
function( M, R )
local c, s;
c := NrColumns( M );
s := List( [ 1 .. NrRows( M ) ], i -> List( [ 1 .. c ], j -> MatElmAsString( M, i, j ) ) );
s := JoinStringsWithSeparator( List( s, JoinStringsWithSeparator ), "],[" );
return Concatenation( "[[", s, "]]" );
end );
##
InstallMethod( GetListListOfHomalgMatrixAsString,
"for homalg internal matrices",
[ IsHomalgInternalMatrixRep, IsHomalgInternalRingRep ],
function( M, R )
local s;
s := GetListListOfStringsOfHomalgMatrix( M, R );
s := List( List( s, JoinStringsWithSeparator ), r -> Concatenation( "[", r, "]" ) );
s := JoinStringsWithSeparator( s );
return Concatenation( "[", s, "]" );
end );
##
InstallMethod( GetSparseListOfHomalgMatrixAsString,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return GetSparseListOfHomalgMatrixAsString( M, HomalgRing( M ) );
end );
##
InstallMethod( GetSparseListOfHomalgMatrixAsString,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgRing ],
function( M, R )
local c, s, i, j, e;
c := NrColumns( M );
s := [ ];
for i in [ 1 .. NrRows( M ) ] do
for j in [ 1 .. c ] do
e := MatElm( M, i, j );
if not IsZero( e ) then
Add( s, [ String( i ), String( j ), String( e ) ] );
fi;
od;
od;
s := JoinStringsWithSeparator( List( s, JoinStringsWithSeparator ), "],[" );
return Concatenation( "[[", s, "]]" );
end );
##
InstallMethod( GetSparseListOfHomalgMatrixAsString,
"for homalg internal matrices",
[ IsHomalgInternalMatrixRep, IsHomalgInternalRingRep ],
function( M, R )
local s, m, r, c, z;
if not IsInternalMatrixHull( Eval( M ) ) then
TryNextMethod( );
fi;
s := Eval( M )!.matrix;
if HasCharacteristic( R ) then
m := Characteristic( R );
if m > 0 and not HasCoefficientsRing( R ) then ## FIXME: we can only deal with Z/mZ and GF(p): m = Size( R ) !!!
if IsPrime( m ) then
s := List( s, a -> List( a, IntFFE ) );
else
s := List( s, a -> List( a, b -> b![1] ) );
fi;
fi;
fi;
r := NrRows( M );
c := NrColumns( M );
z := Zero( R );
s := List( [ 1 .. r ], a -> Filtered( List( [ 1 .. c ], function( b ) if s[a][b] <> z then return [ a, b, s[a][b] ]; else return 0; fi; end ), x -> x <> 0 ) );
s := Concatenation( s );
s := String( s );
RemoveCharacters( s, "\\\n " );
return s;
end );
##
InstallMethod( EntriesOfHomalgMatrixAsListList,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local cols;
cols := [ 1 .. NrColumns( M ) ];
return List( [ 1 .. NrRows( M ) ], r -> List( cols, c -> MatElm( M, r, c ) ) );
end );
##
InstallMethod( EntriesOfHomalgMatrix,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return Flat( EntriesOfHomalgMatrixAsListList( M ) );
end );
##
InstallMethod( GetUnitPosition,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return GetUnitPosition( M, [ ] );
end );
##
InstallMethod( GetCleanRowsPositions,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return GetCleanRowsPositions( M, [ 1 .. NrColumns( M ) ] );
end );
##
InstallMethod( GetColumnIndependentUnitPositions,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return GetColumnIndependentUnitPositions( M, [ ] );
end );
##
InstallMethod( GetRowIndependentUnitPositions,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return GetRowIndependentUnitPositions( M, [ ] );
end );
##
InstallMethod( AreComparableMatrices,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( M1, M2 )
if HasNrRows( M1 ) or HasNrRows( M2 ) then ## trigger as few as possible operations
return IsIdenticalObj( HomalgRing( M1 ), HomalgRing( M2 ) )
and NrRows( M1 ) = NrRows( M2 ) and NrColumns( M1 ) = NrColumns( M2 );
else ## no other choice
return IsIdenticalObj( HomalgRing( M1 ), HomalgRing( M2 ) )
and NrColumns( M1 ) = NrColumns( M2 ) and NrRows( M1 ) = NrRows( M2 );
fi;
end );
##
InstallMethod( \=,
"for internal matrix hulls",
[ IsInternalMatrixHull, IsInternalMatrixHull ],
function( M1, M2 )
if M1!.matrix = M2!.matrix then
return true;
fi;
return false;
end );
## <#GAPDoc Label="EQ:matrix">
## <ManSection>
## <Oper Arg="A, B" Name="\=" Label="for matrices"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; matrices <A>A</A> and <A>B</A> are equal (enter: <A>A</A> <C>=</C> <A>B</A>;),
## taking possible ring relations into account.<P/>
## (for the installed standard method see <Ref Meth="AreEqualMatrices" Label="homalgTable entry"/>)
## <Example><![CDATA[
## gap> ZZ := HomalgRingOfIntegers( );
## Z
## gap> A := HomalgMatrix( "[ 1 ]", ZZ );
## <A 1 x 1 matrix over an internal ring>
## gap> B := HomalgMatrix( "[ 3 ]", ZZ );
## <A 1 x 1 matrix over an internal ring>
## gap> Z2 := ZZ / 2;
## Z/( 2 )
## gap> A := Z2 * A;
## <A 1 x 1 matrix over a residue class ring>
## gap> B := Z2 * B;
## <A 1 x 1 matrix over a residue class ring>
## gap> Display( A );
## [ [ 1 ] ]
##
## modulo [ 2 ]
## gap> Display( B );
## [ [ 3 ] ]
##
## modulo [ 2 ]
## gap> A = B;
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( \=,
"for homalg comparable matrices",
[ IsHomalgMatrix, IsHomalgMatrix ], 10001,
function( M1, M2 )
if not AreComparableMatrices( M1, M2 ) then
return false;
fi;
TryNextMethod( );
end );
##
InstallMethod( \=,
"for homalg comparable internal matrices",
[ IsHomalgInternalMatrixRep, IsHomalgInternalMatrixRep ],
function( M1, M2 )
local RP;
RP := homalgTable( HomalgRing( M1 ) );
if IsBound( RP!.AreEqualMatrices ) then
if RP!.AreEqualMatrices( M1, M2 ) then
## do not touch mutable matrices
if not ( IsMutable( M1 ) or IsMutable( M2 ) ) then
MatchPropertiesAndAttributes( M1, M2,
LIMAT.intrinsic_properties,
LIMAT.intrinsic_attributes,
LIMAT.intrinsic_components
);
fi;
return true;
fi;
elif Eval( M1 ) = Eval( M2 ) then
## do not touch mutable matrices
if not ( IsMutable( M1 ) or IsMutable( M2 ) ) then
MatchPropertiesAndAttributes( M1, M2,
LIMAT.intrinsic_properties,
LIMAT.intrinsic_attributes,
LIMAT.intrinsic_components
);
fi;
return true;
fi;
return false;
end );
##
InstallMethod( ZeroMutable,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return HomalgZeroMatrix( NrRows( M ), NrColumns( M ), HomalgRing( M ) );
end );
## <#GAPDoc Label="Involution">
## <ManSection>
## <Meth Arg="M" Name="Involution" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## The twisted transpose of the &homalg; matrix <A>M</A>.<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with Involution"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( Involution,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local C;
C := HomalgMatrixWithAttributes( [
EvalInvolution, M,
NrRows, NrColumns( M ),
NrColumns, NrRows( M ),
], HomalgRing( M ) );
SetItsInvolution( M, C );
return C;
end );
##
InstallMethod( Involution,
"for homalg matrices",
[ IsHomalgMatrix and HasItsInvolution ],
function( M )
return ItsInvolution( M );
end );
## <#GAPDoc Label="CertainRows">
## <ManSection>
## <Meth Arg="M, plist" Name="CertainRows" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## The matrix of which the <M>i</M>-th row is the <M>k</M>-th row of the &homalg; matrix <A>M</A>,
## where <M>k=</M><A>plist</A><M>[i]</M>.<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with CertainRows"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( CertainRows,
"for homalg matrices",
[ IsHomalgMatrix, IsList ],
function( M, plist )
return HomalgMatrixWithAttributes( [
EvalCertainRows, [ M, plist ],
NrRows, Length( plist ),
NrColumns, NrColumns( M )
], HomalgRing( M ) );
end );
## <#GAPDoc Label="CertainColumns">
## <ManSection>
## <Meth Arg="M, plist" Name="CertainColumns" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## The matrix of which the <M>j</M>-th column is the <M>l</M>-th column of the &homalg; matrix <A>M</A>,
## where <M>l=</M><A>plist</A><M>[i]</M>.<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with CertainColumns"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( CertainColumns,
"for homalg matrices",
[ IsHomalgMatrix, IsList ],
function( M, plist )
return HomalgMatrixWithAttributes( [
EvalCertainColumns, [ M, plist ],
NrColumns, Length( plist ),
NrRows, NrRows( M )
], HomalgRing( M ) );
end );
## <#GAPDoc Label="UnionOfRows">
## <ManSection>
## <Meth Arg="A, B" Name="UnionOfRows" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Stack the two &homalg; matrices <A>A</A> and <A>B</A>.<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with UnionOfRows"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( UnionOfRows,
"of two homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( A, B )
return HomalgMatrixWithAttributes( [
EvalUnionOfRows, [ A, B ],
NrRows, NrRows( A ) + NrRows( B ),
NrColumns, NrColumns( A )
], HomalgRing( A ) );
end );
## <#GAPDoc Label="UnionOfColumns">
## <ManSection>
## <Meth Arg="A, B" Name="UnionOfColumns" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Augment the two &homalg; matrices <A>A</A> and <A>B</A>.<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with UnionOfColumns"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( UnionOfColumns,
"of two homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( A, B )
return HomalgMatrixWithAttributes( [
EvalUnionOfColumns, [ A, B ],
NrRows, NrRows( A ),
NrColumns, NrColumns( A ) + NrColumns( B )
], HomalgRing( A ) );
end );
## <#GAPDoc Label="DiagMat">
## <ManSection>
## <Meth Arg="list" Name="DiagMat" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Build the block diagonal matrix out of the &homalg; matrices listed in <A>list</A>.
## An error is issued if <A>list</A> is empty or if one of the arguments is not a &homalg; matrix.<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with DiagMat"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( DiagMat,
"of two homalg matrices",
[ IsHomogeneousList ],
function( list )
if IsEmpty( list ) then
Error( "the given list of diagonal blocks is empty\n" );
elif not ForAll( list, IsHomalgMatrix ) then
Error( "expected a list of homalg matrices\n" );
fi;
return HomalgMatrixWithAttributes( [
EvalDiagMat, list,
NrRows, Sum( List( list, NrRows ) ),
NrColumns, Sum( List( list, NrColumns ) )
], HomalgRing( list[1] ) );
end );
## <#GAPDoc Label="KroneckerMat">
## <ManSection>
## <Meth Arg="A, B" Name="KroneckerMat" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## The Kronecker (or tensor) product of the two &homalg; matrices <A>A</A> and <A>B</A>.<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with KroneckerMat"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( KroneckerMat,
"of two homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( A, B )
return HomalgMatrixWithAttributes( [
EvalKroneckerMat, [ A, B ],
NrRows, NrRows( A ) * NrRows( B ),
NrColumns, NrColumns( A ) * NrColumns ( B )
], HomalgRing( A ) );
end );
##
InstallMethod( \*,
"for internal matrix hulls",
[ IsRingElement, IsInternalMatrixHull ], 1001, ## it could otherwise run into the method ``PROD: negative integer * additive element with inverse'', value: 24
function( a, A )
return homalgInternalMatrixHull( a * A!.matrix );
end );
## <#GAPDoc Label="MulMat">
## <ManSection>
## <Meth Arg="a, A" Name="\*" Label="for ring elements and matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## The product of the ring element <A>a</A> with the &homalg; matrix <A>A</A> (enter: <A>a</A> <C>*</C> <A>A</A>;).<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with MulMat"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( \*,
"for a homalg matrix and a homalg ring element",
[ IsHomalgMatrix, IsRingElement ], 1001, ## it could otherwise run into the method ``PROD: negative integer * additive element with inverse'', value: 24 (if this value is increased, the corresonding values for \* in LIMAT, COLEM, and below must be increased as well!!!)
function( A, a )
return HomalgMatrixWithAttributes( [
EvalMulMatRight, [ A, a ],
NrRows, NrRows( A ),
NrColumns, NrColumns( A )
], HomalgRing( A ) );
end );
##
InstallMethod( \*,
"for a homalg ring element and a homalg matrix",
[ IsRingElement, IsHomalgMatrix ], 1001, ## it could otherwise run into the method ``PROD: negative integer * additive element with inverse'', value: 24 (if this value is increased, the corresonding values for \* in LIMAT, COLEM, and below must be increased as well!!!)
function( a, A )
return HomalgMatrixWithAttributes( [
EvalMulMat, [ a, A ],
NrRows, NrRows( A ),
NrColumns, NrColumns( A )
], HomalgRing( A ) );
end );
##
InstallMethod( \+,
"for pairs of internal matrix hulls",
[ IsInternalMatrixHull, IsInternalMatrixHull ],
function( A, B )
return homalgInternalMatrixHull( A!.matrix + B!.matrix );
end );
## <#GAPDoc Label="AddMat">
## <ManSection>
## <Meth Arg="A, B" Name="\+" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## The sum of the two &homalg; matrices <A>A</A> and <A>B</A> (enter: <A>A</A> <C>+</C> <A>B</A>;).<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with AddMat"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( \+,
"for pairs of homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( A, B )
return HomalgMatrixWithAttributes( [
EvalAddMat, [ A, B ],
NrRows, NrRows( A ),
NrColumns, NrColumns( A )
], HomalgRing( A ) );
end );
## a synonym of `-<elm>':
InstallMethod( AdditiveInverseMutable,
"for internal matrix hulls",
[ IsInternalMatrixHull ],
function( A )
return homalgInternalMatrixHull( -A!.matrix );
end );
## a synonym of `-<elm>':
InstallMethod( AdditiveInverseMutable,
"for homalg matrices",
[ IsHomalgMatrix ],
function( A )
local R, C;
R := HomalgRing( A );
C := MinusOne( R ) * A;
if HasIsZero( A ) then
SetIsZero( C, IsZero( A ) );
fi;
return C;
end );
## a synonym of `-<elm>':
InstallMethod( AdditiveInverseMutable,
"for homalg matrices",
[ IsHomalgMatrix and IsZero ],
function( A )
return A;
end );
##
InstallMethod( \-,
"for pairs of internal matrix hulls",
[ IsInternalMatrixHull, IsInternalMatrixHull ],
function( A, B )
return homalgInternalMatrixHull( A!.matrix - B!.matrix );
end );
## <#GAPDoc Label="SubMat">
## <ManSection>
## <Meth Arg="A, B" Name="\-" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## The difference of the two &homalg; matrices <A>A</A> and <A>B</A> (enter: <A>A</A> <C>-</C> <A>B</A>;).<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with SubMat"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( \-,
"for pairs of of homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( A, B )
return HomalgMatrixWithAttributes( [
EvalSubMat, [ A, B ],
NrRows, NrRows( A ),
NrColumns, NrColumns( A )
], HomalgRing( A ) );
end );
##
InstallMethod( \*,
"for pairs of internal matrix hulls",
[ IsInternalMatrixHull, IsInternalMatrixHull ],
function( A, B )
return homalgInternalMatrixHull( A!.matrix * B!.matrix );
end );
## <#GAPDoc Label="Compose:matrix">
## <ManSection>
## <Meth Arg="A, B" Name="\*" Label="for composable matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## The matrix product of the two &homalg; matrices <A>A</A> and <A>B</A> (enter: <A>A</A> <C>*</C> <A>B</A>;).<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with Compose"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( \*,
"for pairs of homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ], 14001, ## it could otherwise run into the method ``PROD: IsRingElement * IsHomalgMatrix'', value: 1001 (if this value is increased, the corresonding values in LIMAT must be increased as well!!!)
function( A, B )
return HomalgMatrixWithAttributes( [
EvalCompose, [ A, B ],
NrRows, NrRows( A ),
NrColumns, NrColumns( B )
], HomalgRing( A ) );
end );
##
InstallMethod( POW,
"for homalg maps",
[ IsHomalgMatrix, IsInt ],
function( A, pow )
local R;
if NrRows( A ) <> NrColumns( A ) then
Error( "the matrix is not quadratic\n" );
fi;
R := HomalgRing( A );
if pow < 0 then
Error( "not implemented yet\n" );
elif pow = 0 then
return HomalgIdentityMatrix( NrRows( A ), R );
elif pow = 1 then
return A;
else
return Iterated( ListWithIdenticalEntries( pow, A ), \* );
fi;
end );
##
InstallMethod( NonZeroRows,
"for homalg matrices",
[ IsHomalgMatrix ],
function( C )
local zero_rows;
zero_rows := ZeroRows( C );
return Filtered( [ 1 .. NrRows( C ) ], x -> not x in zero_rows );
end );
##
InstallMethod( NonZeroColumns,
"for homalg matrices",
[ IsHomalgMatrix ],
function( C )
local zero_columns;
zero_columns := ZeroColumns( C );
return Filtered( [ 1 .. NrColumns( C ) ], x -> not x in zero_columns );
end );
##
InstallMethod( AdjunctMatrix,
"for homalg matrices",
[ IsHomalgMatrix ],
function( C )
local R, m, A;
R := HomalgRing( C );
if not HasIsCommutative( R ) then
Error( "the ring is not known to be commutative\n" );
elif not IsCommutative( R ) then
Error( "the ring is not commutative\n" );
fi;
m := NrRows( C );
if not m = NrColumns( C ) then
Error( "the input ", m, "x", NrColumns( C ), "-matrix is not quadratic\n" );
fi;
m := [ 1 .. m ];
A := List( m, c -> List( m, r -> (-1)^(r+c) * Determinant( CertainRows( CertainColumns( C, Difference( m, [ c ] ) ), Difference( m, [ r ] ) ) ) ) );
return HomalgMatrix( A, R );
end );
## <#GAPDoc Label="LeftInverseLazy">
## <ManSection>
## <Oper Arg="M" Name="LeftInverseLazy" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## A lazy evaluated left inverse <M>C</M> of the matrix <A>M</A>. If no left inverse exists then
## <C>Eval</C>( <A>C</A> ) will issue an error.<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with LeftInverseLazy"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( LeftInverseLazy,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local C;
## we assume the LeftInverse exists
C := HomalgMatrixWithAttributes( [
EvalLeftInverse, M,
NrRows, NrColumns( M ),
NrColumns, NrRows( M )
], HomalgRing( M ) );
## check assertion
Assert( 6, not IsBool( Eval( C ) ) );
## SetLeftInverse( M, C ) will cause a infinite loop
return C;
end );
## <#GAPDoc Label="RightInverseLazy">
## <ManSection>
## <Oper Arg="M" Name="RightInverseLazy" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## A lazy evaluated right inverse <M>C</M> of the matrix <A>M</A>. If no right inverse exists then
## <C>Eval</C>( <A>C</A> ) will issue an error.<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with RightInverseLazy"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( RightInverseLazy,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local C;
## we assume the RightInverse exists
C := HomalgMatrixWithAttributes( [
EvalRightInverse, M,
NrColumns, NrRows( M ),
NrRows, NrColumns( M )
], HomalgRing( M ) );
## check assertion
Assert( 6, not IsBool( Eval( C ) ) );
## SetRightInverse( M, C ) will cause a infinite loop
return C;
end );
##
InstallMethod( DiagonalEntries,
"of homalg matrices",
[ IsHomalgMatrix ],
function( M )
local m;
m := Minimum( NrRows( M ), NrColumns( M ) );
return List( [ 1 .. m ], a -> MatElm( M, a, a ) );
end );
##
InstallMethod( Minors,
"of homalg matrices",
[ IsInt, IsHomalgMatrix ],
function( d, M )
local R, r, c, l;
R := HomalgRing( M );
if not HasIsCommutative( R ) then
Error( "the ring is not known to be commutative\n" );
elif not IsCommutative( R ) then
Error( "the ring is not commutative\n" );
fi;
if d <= 0 then
return [ One( R ) ];
fi;
r := NrRows( M );
c := NrColumns( M );
if d > Minimum( r, c ) then
return [ Zero( R ) ];
fi;
l := Cartesian( Combinations( [ 1 .. r ], d ), Combinations( [ 1 .. c ], d ) );
l := List( l, rc -> Determinant( CertainColumns( CertainRows( M, rc[1] ), rc[2] ) ) );
l := Filtered( l, m -> not IsZero( m ) );
if l = [ ] then
return [ Zero( R ) ];
fi;
return DuplicateFreeList( l );
end );
##
InstallMethod( MaximalMinors,
"of homalg matrices",
[ IsHomalgMatrix ],
function( M )
return Minors( Minimum( NrRows( M ), NrColumns( M ) ), M );
end );
##
InstallMethod( PostMakeImmutable,
"for homalg internal matrices",
[ IsHomalgInternalMatrixRep and HasEval ],
function( A )
MakeImmutable( Eval( A )!.matrix );
end );
##
InstallMethod( SetIsMutableMatrix,
"for homalg matrices and a Boolean",
[ IsHomalgMatrix, IsBool ],
function( M, b )
if b = true then;
SetFilterObj( M, IsMutable );
else
ResetFilterObj( M, IsMutable );
fi;
end );
##
InstallMethod( SetIsMutableMatrix,
"for homalg matrices and a Boolean",
[ IsHomalgMatrix and IsEmptyMatrix, IsBool ], 1001,
function( M, b )
## do nothing
end );
##
InstallMethod( Iterator,
"of homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, c, d, rank, iter, save, F, r;
R := HomalgRing( M );
if not IsFieldForHomalg( R ) or
( not HasCharacteristic( R ) or Characteristic( R ) = 0 ) or
( not HasDegreeOverPrimeField( R ) or not IsInt( DegreeOverPrimeField( R ) ) ) then
TryNextMethod( );
fi;
c := Characteristic( R );
d := DegreeOverPrimeField( R );
M := BasisOfRows( M );
if not IsLeftRegular( M ) then
TryNextMethod( );
fi;
rank := RowRankOfMatrix( M );
iter := Iterator( GF(c^d)^rank );
if IsHomalgInternalRingRep( R ) then
F := R;
else
F := HomalgRingOfIntegers( c, d );
fi;
r := rec(
ring := R,
iter := iter,
matrix := M,
rank := rank,
GF := F,
NextIterator :=
function( i )
local mat;
mat := HomalgMatrix( NextIterator( i!.iter ), 1, i!.rank, i!.GF );
SetNrRows( mat, 1 ); ## should be obsolete
SetNrColumns( mat, i!.rank ); ## should be obsolete
return ( i!.ring * mat ) * i!.matrix;
end,
IsDoneIterator :=
function( i )
return IsDoneIterator( i!.iter );
end,
ShallowCopy :=
function( i );
return
rec(
ring := i!.ring,
iter := ShallowCopy( i!.iter ),
matrix := i!.matrix,
rank := i!.rank,
GF := i!.GF,
NextIterator := i!.NextIterator,
IsDoneIterator := i!.IsDoneIterator,
ShallowCopy := i!.ShallowCopy
);
end
);
return IteratorByFunctions( r );
end );
##
InstallMethod( Select,
"for a matrix and a list",
[ IsHomalgMatrix, IsList ],
function( M, L )
local R, indets, zero, map, N;
R := HomalgRing( M );
if HasRelativeIndeterminatesOfPolynomialRing( R ) then
indets := RelativeIndeterminatesOfPolynomialRing( R );
elif HasIndeterminatesOfPolynomialRing( R ) then
indets := IndeterminatesOfPolynomialRing( R );
else
TryNextMethod( );
fi;
if not IsSubset( indets, L ) then
Error( "the second argument is not a subset of the list of indeterminates\n" );
fi;
zero := Zero( R );
map := List( indets, function( a ) if a in L then return a; else return zero; fi; end );
map := RingMap( map, R, R );
N := Pullback( map, M );
return CertainRows( M, ZeroRows( M - N ) );
end );
####################################
#
# constructor functions and methods:
#
####################################
##
InstallGlobalFunction( homalgInternalMatrixHull,
function( M )
return Objectify( TheTypeInternalMatrixHull, rec( matrix := M ) );
end );
##
InstallMethod( ConvertHomalgMatrixViaListListString,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgRing ],
function( M, R )
local s;
s := GetListListOfHomalgMatrixAsString( M );
return CreateHomalgMatrixFromString( s, R );
end );
##
InstallMethod( ConvertHomalgMatrixViaListListString,
"for homalg matrices",
[ IsHomalgMatrix, IsInt, IsInt, IsHomalgRing ],
function( M, r, c, R )
local s;
s := GetListListOfHomalgMatrixAsString( M );
return CreateHomalgMatrixFromString( s, R );
end );
##
InstallMethod( ConvertHomalgMatrixViaListString,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgRing ],
function( M, R )
local r, c, s;
r := NrRows( M );
c := NrColumns( M );
s := GetListOfHomalgMatrixAsString( M );
return CreateHomalgMatrixFromString( s, r, c, R );
end );
##
InstallMethod( ConvertHomalgMatrixViaListString,
"for homalg matrices",
[ IsHomalgMatrix, IsInt, IsInt, IsHomalgRing ],
function( M, r, c, R )
local s;
s := GetListOfHomalgMatrixAsString( M );
return CreateHomalgMatrixFromString( s, r, c, R );
end );
##
InstallMethod( ConvertHomalgMatrixViaSparseString,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgRing ],
function( M, R )
local r, c, s;
r := NrRows( M );
c := NrColumns( M );
s := GetSparseListOfHomalgMatrixAsString( M );
return CreateHomalgMatrixFromSparseString( s, r, c, R );
end );
##
InstallMethod( ConvertHomalgMatrixViaSparseString,
"for homalg matrices",
[ IsHomalgMatrix, IsInt, IsInt, IsHomalgRing ],
function( M, r, c, R )
local s;
s := GetSparseListOfHomalgMatrixAsString( M );
return CreateHomalgMatrixFromSparseString( s, r, c, R );
end );
##
InstallMethod( ConvertHomalgMatrix,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgRing ],
function( M, R )
if IsBound( M!.ConvertHomalgMatrixViaSparseString ) and M!.ConvertHomalgMatrixViaSparseString = true then
return ConvertHomalgMatrixViaSparseString( M, R );
fi;
return ConvertHomalgMatrixViaListListString( M, R );
end );
##
InstallMethod( ConvertHomalgMatrix,
"for homalg matrices",
[ IsHomalgMatrix, IsInt, IsInt, IsHomalgRing ],
function( M, r, c, R )
if IsBound( M!.ConvertHomalgMatrixViaSparseString ) and M!.ConvertHomalgMatrixViaSparseString = true then
return ConvertHomalgMatrixViaSparseString( M, r, c, R );
fi;
return ConvertHomalgMatrixViaListString( M, r, c, R );
end );
##
InstallMethod( CreateHomalgMatrixFromString,
"constructor for homalg matrices",
[ IsString, IsHomalgInternalRingRep ],
function( S, R )
local s;
s := ShallowCopy( S );
RemoveCharacters( s, "\\\n\" " );
return HomalgMatrix( EvalString( s ), R );
end );
##
InstallMethod( CreateHomalgMatrixFromString,
"constructor for homalg matrices",
[ IsString, IsInt, IsInt, IsHomalgInternalRingRep ],
function( S, r, c, R )
local s;
s := ShallowCopy( S );
RemoveCharacters( s, "\\\n\" " );
s := EvalString( s );
if IsMatrix( s ) then
return HomalgMatrix( s, r, c, R );
elif IsHomogeneousList( s ) then
return HomalgMatrix( ListToListList( s, r, c ), r, c, R );
else
Error( "the evaluated string is not in {IsMatrix, IsHomogeneousList}\n" );
fi;
end );
##
InstallMethod( CreateHomalgMatrixFromSparseString,
"constructor for homalg matrices",
[ IsString, IsInt, IsInt, IsHomalgRing ],
function( S, r, c, R )
local s, M, e;
s := ShallowCopy( S );
RemoveCharacters( s, "[]\\\n\" " );
M := HomalgInitialMatrix( r, c, R );
s := SplitString( s, "," );
s := ListToListList( s, Length( s ) / 3, 3 );
Perform( s, function( a ) SetMatElm( M, Int( a[1] ), Int( a[2] ), a[3], R ); end );
ResetFilterObj( M, IsMutable );
return M;
end );
##
InstallMethod( CreateHomalgMatrixFromSparseString,
"constructor for homalg matrices",
[ IsString, IsInt, IsInt, IsHomalgInternalRingRep ],
function( S, r, c, R )
local s, M, e;
s := ShallowCopy( S );
RemoveCharacters( s, "\\\n\" " );
M := List( [ 1 .. r ], a -> List( [ 1 .. c ], b -> Zero( R ) ) );
for e in EvalString( s ) do
M[e[1]][e[2]] := e[3];
od;
return HomalgMatrix( M, r, c, R );
end );
##
InstallMethod( CreateHomalgMatrixFromList,
"constructor for homalg matrices",
[ IsList, IsHomalgRing ],
function( L, R )
local M;
if IsList( L[1] ) then
M := List( L, r -> List( r, String ) );
M := Concatenation( "[[", JoinStringsWithSeparator( List( M, r -> JoinStringsWithSeparator( r ) ), "],[" ), "]]" );
else
## this resembles NormalizeInput in Maple's homalg ( a legacy ;) )
M := Concatenation( "[[", JoinStringsWithSeparator( List( L, String ), "],[" ), "]]" );
## What is the use case for this? Wouldn't it be better to replace this by an error message?
# Error( "the number of rows and columns must be specified to construct a matrix from a list" );
fi;
return CreateHomalgMatrixFromString( M, R );
end );
##
InstallMethod( CreateHomalgMatrixFromList,
"constructor for homalg matrices",
[ IsList, IsInt, IsInt, IsHomalgRing ],
function( L, r, c, R )
local M;
if IsList( L[1] ) then
M := List( Concatenation( L ), String );
M := Concatenation( "[", JoinStringsWithSeparator( M ), "]" );
else
M := Concatenation( "[", JoinStringsWithSeparator( List( L, String ) ), "]" );
fi;
return CreateHomalgMatrixFromString( M, r, c, R );
end );
## <#GAPDoc Label="HomalgMatrix">
## <ManSection>
## <Func Arg="llist, R" Name="HomalgMatrix" Label="constructor for matrices using a listlist"/>
## <Func Arg="list, m, n, R" Name="HomalgMatrix" Label="constructor for matrices using a list"/>
## <Func Arg="str_llist, R" Name="HomalgMatrix" Label="constructor for matrices using a string of a listlist"/>
## <Func Arg="str_list, m, n, R" Name="HomalgMatrix" Label="constructor for matrices using a string of a list"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## An immutable evaluated <M><A>m</A> \times <A>n</A></M> &homalg; matrix over the &homalg; ring <A>R</A>.
## <Example><![CDATA[
## gap> ZZ := HomalgRingOfIntegers( );
## Z
## gap> m := HomalgMatrix( [ [ 1, 2, 3 ], [ 4, 5, 6 ] ], ZZ );
## <A 2 x 3 matrix over an internal ring>
## gap> Display( m );
## [ [ 1, 2, 3 ],
## [ 4, 5, 6 ] ]
## ]]></Example>
## <Example><![CDATA[
## gap> m := HomalgMatrix( [ [ 1, 2, 3 ], [ 4, 5, 6 ] ], 2, 3, ZZ );
## <A 2 x 3 matrix over an internal ring>
## gap> Display( m );
## [ [ 1, 2, 3 ],
## [ 4, 5, 6 ] ]
## ]]></Example>
## <Example><![CDATA[
## gap> m := HomalgMatrix( [ 1, 2, 3, 4, 5, 6 ], 2, 3, ZZ );
## <A 2 x 3 matrix over an internal ring>
## gap> Display( m );
## [ [ 1, 2, 3 ],
## [ 4, 5, 6 ] ]
## ]]></Example>
## <Example><![CDATA[
## gap> m := HomalgMatrix( "[ [ 1, 2, 3 ], [ 4, 5, 6 ] ]", ZZ );
## <A 2 x 3 matrix over an internal ring>
## gap> Display( m );
## [ [ 1, 2, 3 ],
## [ 4, 5, 6 ] ]
## ]]></Example>
## <Example><![CDATA[
## gap> m := HomalgMatrix( "[ [ 1, 2, 3 ], [ 4, 5, 6 ] ]", 2, 3, ZZ );
## <A 2 x 3 matrix over an internal ring>
## gap> Display( m );
## [ [ 1, 2, 3 ],
## [ 4, 5, 6 ] ]
## ]]></Example>
## It is nevertheless recommended to use the following form to create &homalg; matrices. This
## form can also be used to define external matrices. Since whitespaces
## (&see; <Ref Label="Whitespaces" BookName="Ref"/>) are ignored,
## they can be used as optical delimiters:
## <Example><![CDATA[
## gap> m := HomalgMatrix( "[ 1, 2, 3, 4, 5, 6 ]", 2, 3, ZZ );
## <A 2 x 3 matrix over an internal ring>
## gap> Display( m );
## [ [ 1, 2, 3 ],
## [ 4, 5, 6 ] ]
## ]]></Example>
## One can split the input string over several lines using the backslash character '\' to end each line
## <Example><![CDATA[
## gap> m := HomalgMatrix( "[ \
## > 1, 2, 3, \
## > 4, 5, 6 \
## > ]", 2, 3, ZZ );
## <A 2 x 3 matrix over an internal ring>
## gap> Display( m );
## [ [ 1, 2, 3 ],
## [ 4, 5, 6 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgMatrix,
function( arg )
local nargs, M, R, RP, type, matrix, nr_rows, nr_columns;
nargs := Length( arg );
M := arg[1];
R := arg[nargs];
if nargs > 1 and M <> [ ] then
if HasConstructorForHomalgMatrices( R ) then
return CallFuncList( ConstructorForHomalgMatrices( R ), arg );
fi;
if IsString( M ) then
return CallFuncList( CreateHomalgMatrixFromString, arg );
elif not IsHomalgInternalRingRep( R ) and ## the ring R is not internal,
( IsList( M ) or ## while M is either a list of ring elements,
IsMatrix( M ) ) then ## or a matrix of (hopefully) ring elements
return CallFuncList( CreateHomalgMatrixFromList, arg );
fi;
fi;
if not IsHomalgRing( R ) then
Error( "the last argument must be an IsHomalgRing\n" );
fi;
## here we take care of the degenerate input M = [ ] for all rings
if M = [ ] then ## CreateHomalgMatrixFromString also covers M = "", "[]", etc.
if nargs = 2 then
return HomalgZeroMatrix( 0, 0, R );
else
return CallFuncList( HomalgZeroMatrix, arg{[ 2 .. nargs ]} );
fi;
fi;
if HasTypeOfHomalgMatrix( R ) then
type := TypeOfHomalgMatrix( R );
elif IsHomalgInternalRingRep( R ) then
type := TheTypeHomalgInternalMatrix;
else
Error( "the homalg ring must contain the type of the matrices as the attribute TypeOfHomalgMatrix\n" );
fi;
matrix := rec( ring := R );
## here we take care of the case when only the ring is given
if nargs = 1 then
## Objectify:
Objectify( type, matrix );
## an empty matrix
return matrix;
fi;
RP := homalgTable( R );
if IsList( M ) and Length( M ) > 0 and not IsList( M[1] ) and
ForAll( M, IsRingElement ) then
if Length( arg ) > 2 and arg[2] in NonnegativeIntegers then
M := ListToListList( M, arg[2], Length( M ) / arg[2] );
else
M := List( M, a -> [ a ] ); ## this resembles NormalizeInput in Maple's homalg ( a legacy ;) )
fi;
if IsBound(RP!.ImportMatrix) then
M := RP!.ImportMatrix( M, R );
fi;
elif IsInternalMatrixHull( M ) then ## why are we doing this? for ShallowCopy?
if IsMatrix( M!.matrix ) then
M := M!.matrix;
else
M := homalgInternalMatrixHull( M!.matrix );
fi;
elif IsMatrix( M ) and IsBound(RP!.ImportMatrix) then
M := RP!.ImportMatrix( M, R );
else
M := ShallowCopy( M ); ## by this we are sure that possible changes to a mutable GAP matrix arg[1] does not destroy the logic of homalg
fi;
if IsHomalgInternalRingRep( R ) and
not IsInternalMatrixHull( M ) then ## TheTypeHomalgInternalMatrix
if IsMatrix( M ) then
## Objectify:
ObjectifyWithAttributes(
matrix, type,
NrRows, Length( M ),
NrColumns, Length( M[1] ),
Eval, homalgInternalMatrixHull( M ) );
elif IsList( M ) then
## Objectify:
ObjectifyWithAttributes(
matrix, type,
Eval, homalgInternalMatrixHull( M ) );
if M = [ ] then
SetNrRows( matrix, 0 );
SetNrColumns( matrix, 0 );
elif M[1] = [] then
SetNrRows( matrix, Length( M ) );
SetNrColumns( matrix, 0 );
fi;
else
if Length( arg ) > 2 and arg[2] in NonnegativeIntegers then
nr_rows := true;
else
nr_rows := false;
fi;
if Length( arg ) > 3 and arg[3] in NonnegativeIntegers then
nr_columns := true;
else
nr_columns := false;
fi;
if nr_rows and nr_columns then
## Objectify:
ObjectifyWithAttributes(
matrix, type,
NrRows, arg[2],
NrColumns, arg[3],
Eval, M );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
Eval, M );
if nr_rows then
SetNrRows( matrix, arg[2] );
fi;
if nr_columns then
SetNrColumns( matrix, arg[3] );
fi;
fi;
fi;
else
if Length( arg ) > 2 and arg[2] in NonnegativeIntegers then
nr_rows := true;
else
nr_rows := false;
fi;
if Length( arg ) > 3 and arg[3] in NonnegativeIntegers then
nr_columns := true;
else
nr_columns := false;
fi;
if nr_rows and nr_columns then
## Objectify:
ObjectifyWithAttributes(
matrix, type,
NrRows, arg[2],
NrColumns, arg[3],
Eval, M );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
Eval, M );
if nr_rows then
SetNrRows( matrix, arg[2] );
fi;
if nr_columns then
SetNrColumns( matrix, arg[3] );
fi;
fi;
fi;
return matrix;
end );
##
InstallGlobalFunction( HomalgMatrixWithAttributes,
function( attributes, R )
local matrix;
## for performance reasons this internal function
## should not perform syntax checks
matrix := rec( ring := R );
## ObjectifyWithAttributes:
CallFuncList( ObjectifyWithAttributes,
Concatenation( [ matrix, TypeOfHomalgMatrix( R ) ], attributes )
);
return matrix;
end );
## <#GAPDoc Label="HomalgZeroMatrix">
## <ManSection>
## <Func Arg="m, n, R" Name="HomalgZeroMatrix" Label="constructor for zero matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## An immutable unevaluated <M><A>m</A> \times <A>n</A></M> &homalg; zero matrix over the &homalg; ring <A>R</A>.
## <Example><![CDATA[
## gap> ZZ := HomalgRingOfIntegers( );
## Z
## gap> z := HomalgZeroMatrix( 2, 3, ZZ );
## <An unevaluated 2 x 3 zero matrix over an internal ring>
## gap> Display( z );
## [ [ 0, 0, 0 ],
## [ 0, 0, 0 ] ]
## gap> z;
## <A 2 x 3 zero matrix over an internal ring>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgZeroMatrix,
function( arg ) ## the zero matrix
local R, type, matrix, nr_rows, nr_columns;
R := arg[Length( arg )];
if not IsHomalgRing( R ) then
Error( "the last argument must be an IsHomalgRing\n" );
fi;
if HasTypeOfHomalgMatrix( R ) then
type := TypeOfHomalgMatrix( R );
elif IsHomalgInternalRingRep( R ) then
type := TheTypeHomalgInternalMatrix;
else
Error( "the homalg ring must contain the type of the matrices as the attribute TypeOfHomalgMatrix\n" );
fi;
matrix := rec( ring := R );
if Length( arg ) > 1 and arg[1] in NonnegativeIntegers then
nr_rows := true;
else
nr_rows := false;
fi;
if Length( arg ) > 2 and arg[2] in NonnegativeIntegers then
nr_columns := true;
else
nr_columns := false;
fi;
if nr_rows and nr_columns then
## cache the 1 x 1 zero matrix
if nr_rows = 1 and nr_columns = 1 then
if IsBound( R!.OneByOneZeroMatrix ) then
return R!.OneByOneZeroMatrix;
else
R!.OneByOneZeroMatrix := matrix;
fi;
fi;
## Objectify:
ObjectifyWithAttributes(
matrix, type,
NrRows, arg[1],
NrColumns, arg[2],
IsZero, true );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
IsZero, true );
if nr_rows then
SetNrRows( matrix, arg[1] );
fi;
if nr_columns then
SetNrColumns( matrix, arg[2] );
fi;
fi;
return matrix;
end );
## <#GAPDoc Label="HomalgIdentityMatrix">
## <ManSection>
## <Func Arg="m, R" Name="HomalgIdentityMatrix" Label="constructor for identity matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## An immutable unevaluated <M><A>m</A> \times <A>m</A></M> &homalg; identity matrix over the &homalg; ring <A>R</A>.
## <Example><![CDATA[
## gap> ZZ := HomalgRingOfIntegers( );
## Z
## gap> id := HomalgIdentityMatrix( 3, ZZ );
## <An unevaluated 3 x 3 identity matrix over an internal ring>
## gap> Display( id );
## [ [ 1, 0, 0 ],
## [ 0, 1, 0 ],
## [ 0, 0, 1 ] ]
## gap> id;
## <A 3 x 3 identity matrix over an internal ring>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgIdentityMatrix,
function( arg ) ## the identity matrix
local R, type, matrix;
R := arg[Length( arg )];
if not IsHomalgRing( R ) then
Error( "the last argument must be an IsHomalgRing\n" );
fi;
if HasTypeOfHomalgMatrix( R ) then
type := TypeOfHomalgMatrix( R );
elif IsHomalgInternalRingRep( R ) then
type := TheTypeHomalgInternalMatrix;
else
Error( "the homalg ring must contain the type of the matrices as the attribute TypeOfHomalgMatrix\n" );
fi;
matrix := rec( ring := R );
if Length( arg ) > 1 and arg[1] in NonnegativeIntegers then
## Objectify:
ObjectifyWithAttributes(
matrix, type,
NrRows, arg[1],
NrColumns, arg[1],
IsOne, true );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
IsOne, true );
fi;
return matrix;
end );
## <#GAPDoc Label="HomalgInitialMatrix">
## <ManSection>
## <Func Arg="m, n, R" Name="HomalgInitialMatrix" Label="constructor for initial matrices filled with zeros"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## A mutable unevaluated initial <M><A>m</A> \times <A>n</A></M> &homalg; matrix filled with zeros
## over the &homalg; ring <A>R</A>. This construction is useful in case one wants to define a matrix
## by assigning its nonzero entries.
## The property <Ref Prop="IsInitialMatrix"/> is reset as soon as the matrix is evaluated.
## New computed properties or attributes of the matrix won't be cached,
## until the matrix is explicitly made immutable using (&see; <Ref Func="MakeImmutable"
## BookName="Reference" Style="Number"/>).
## <Example><![CDATA[
## gap> ZZ := HomalgRingOfIntegers( );
## Z
## gap> z := HomalgInitialMatrix( 2, 3, ZZ );
## <An initial 2 x 3 matrix over an internal ring>
## gap> HasIsZero( z );
## false
## gap> IsZero( z );
## true
## gap> z;
## <A 2 x 3 mutable matrix over an internal ring>
## gap> HasIsZero( z );
## false
## ]]></Example>
## <Example><![CDATA[
## gap> n := HomalgInitialMatrix( 2, 3, ZZ );
## <An initial 2 x 3 matrix over an internal ring>
## gap> SetMatElm( n, 1, 1, "1" );
## gap> SetMatElm( n, 2, 3, "1" );
## gap> MakeImmutable( n );
## <A 2 x 3 matrix over an internal ring>
## gap> Display( n );
## [ [ 1, 0, 0 ],
## [ 0, 0, 1 ] ]
## gap> IsZero( n );
## false
## gap> n;
## <A non-zero 2 x 3 matrix over an internal ring>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgInitialMatrix,
function( arg ) ## an initial matrix having the flag IsInitialMatrix
local R, type, matrix, nr_rows, nr_columns; ## and filled with zeros BUT NOT marked as an IsZero
R := arg[Length( arg )];
if not IsHomalgRing( R ) then
Error( "the last argument must be an IsHomalgRing\n" );
fi;
if HasTypeOfHomalgMatrix( R ) then
type := TypeOfHomalgMatrix( R );
elif IsHomalgInternalRingRep( R ) then
type := TheTypeHomalgInternalMatrix;
else
Error( "the homalg ring must contain the type of the matrices as the attribute TypeOfHomalgMatrix\n" );
fi;
matrix := rec( ring := R );
if Length( arg ) > 1 and arg[1] in NonnegativeIntegers then
nr_rows := true;
else
nr_rows := false;
fi;
if Length( arg ) > 2 and arg[2] in NonnegativeIntegers then
nr_columns := true;
else
nr_columns := false;
fi;
if nr_rows and nr_columns then
## Objectify:
ObjectifyWithAttributes(
matrix, type,
NrRows, arg[1],
NrColumns, arg[2],
IsInitialMatrix, true );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
IsInitialMatrix, true );
if nr_rows then
SetNrRows( matrix, arg[1] );
fi;
if nr_columns then
SetNrColumns( matrix, arg[2] );
fi;
fi;
SetIsMutableMatrix( matrix, true );
return matrix;
end );
## <#GAPDoc Label="HomalgInitialIdentityMatrix">
## <ManSection>
## <Func Arg="m, R" Name="HomalgInitialIdentityMatrix" Label="constructor for initial quadratic matrices with ones on the diagonal"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## A mutable unevaluated initial <M><A>m</A> \times <A>m</A></M> &homalg; quadratic matrix with ones
## on the diagonal over the &homalg; ring <A>R</A>. This construction is useful in case one wants to define
## an elementary matrix by assigning its off-diagonal nonzero entries.
## The property <Ref Prop="IsInitialIdentityMatrix"/> is reset as soon as the matrix is evaluated.
## New computed properties or attributes of the matrix won't be cached,
## until the matrix is explicitly made immutable using (&see; <Ref Func="MakeImmutable"
## BookName="Reference" Style="Number"/>).
## <Example><![CDATA[
## gap> ZZ := HomalgRingOfIntegers( );
## Z
## gap> id := HomalgInitialIdentityMatrix( 3, ZZ );
## <An initial identity 3 x 3 matrix over an internal ring>
## gap> HasIsOne( id );
## false
## gap> IsOne( id );
## true
## gap> id;
## <A 3 x 3 mutable matrix over an internal ring>
## gap> HasIsOne( id );
## false
## ]]></Example>
## <Example><![CDATA[
## gap> e := HomalgInitialIdentityMatrix( 3, ZZ );
## <An initial identity 3 x 3 matrix over an internal ring>
## gap> SetMatElm( e, 1, 2, "1" );
## gap> SetMatElm( e, 2, 1, "-1" );
## gap> MakeImmutable( e );
## <A 3 x 3 matrix over an internal ring>
## gap> Display( e );
## [ [ 1, 1, 0 ],
## [ -1, 1, 0 ],
## [ 0, 0, 1 ] ]
## gap> IsOne( e );
## false
## gap> e;
## <A 3 x 3 matrix over an internal ring>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgInitialIdentityMatrix,
function( arg ) ## a square initial matrix having the flag IsInitialIdentityMatrix
local R, type, matrix; ## and filled with an identity matrix BUT NOT marked as an IsOne
R := arg[Length( arg )];
if not IsHomalgRing( R ) then
Error( "the last argument must be an IsHomalgRing\n" );
fi;
if HasTypeOfHomalgMatrix( R ) then
type := TypeOfHomalgMatrix( R );
elif IsHomalgInternalRingRep( R ) then
type := TheTypeHomalgInternalMatrix;
else
Error( "the homalg ring must contain the type of the matrices as the attribute TypeOfHomalgMatrix\n" );
fi;
matrix := rec( ring := R );
if Length( arg ) > 1 and arg[1] in NonnegativeIntegers then
## Objectify:
ObjectifyWithAttributes(
matrix, type,
NrRows, arg[1],
NrColumns, arg[1],
IsInitialIdentityMatrix, true );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
IsInitialIdentityMatrix, true );
fi;
SetIsMutableMatrix( matrix, true );
return matrix;
end );
## <#GAPDoc Label="HomalgVoidMatrix">
## <ManSection>
## <Func Arg="[ m,][ n,] R" Name="HomalgVoidMatrix" Label="constructor for void matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## A void <M><A>m</A> \times <A>n</A></M> &homalg; matrix.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgVoidMatrix,
function( arg ) ## a void matrix filled with nothing having the flag IsVoidMatrix
local R, type, matrix, nr_rows, nr_columns;
R := arg[Length( arg )];
if not IsHomalgRing( R ) then
Error( "the last argument must be an IsHomalgRing\n" );
fi;
if HasTypeOfHomalgMatrix( R ) then
type := TypeOfHomalgMatrix( R );
elif IsHomalgInternalRingRep( R ) then
type := TheTypeHomalgInternalMatrix;
else
Error( "the homalg ring must contain the type of the matrices as the attribute TypeOfHomalgMatrix\n" );
fi;
matrix := rec( ring := R );
if Length( arg ) > 1 and arg[1] in NonnegativeIntegers then
nr_rows := true;
else
nr_rows := false;
fi;
if Length( arg ) > 2 and arg[2] in NonnegativeIntegers then
nr_columns := true;
else
nr_columns := false;
fi;
if nr_rows and nr_columns then
## Objectify:
ObjectifyWithAttributes(
matrix, type,
NrRows, arg[1],
NrColumns, arg[2],
IsVoidMatrix, true );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
IsVoidMatrix, true );
if nr_rows then
SetNrRows( matrix, arg[1] );
fi;
if nr_columns then
SetNrColumns( matrix, arg[2] );
fi;
fi;
return matrix;
end );
## <#GAPDoc Label="HomalgDiagonalMatrix">
## <ManSection>
## <Func Arg="diag, R" Name="HomalgDiagonalMatrix" Label="constructor for diagonal matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## An immutable unevaluated diagonal &homalg; matrix over the &homalg; ring <A>R</A>. The diagonal
## consists of the entries of the list <A>diag</A>.
## <Example><![CDATA[
## gap> ZZ := HomalgRingOfIntegers( );
## Z
## gap> d := HomalgDiagonalMatrix( [ 1, 2, 3 ], ZZ );
## <An unevaluated diagonal 3 x 3 matrix over an internal ring>
## gap> Display( d );
## [ [ 1, 0, 0 ],
## [ 0, 2, 0 ],
## [ 0, 0, 3 ] ]
## gap> d;
## <A diagonal 3 x 3 matrix over an internal ring>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgDiagonalMatrix,
function( arg ) ## the diagonal matrix
local nargs, R, diag, d, M;
nargs := Length( arg );
if nargs = 0 then
Error( "no arguments provided\n" );
fi;
if IsHomalgRing( arg[nargs] ) then
R := arg[nargs];
fi;
if IsRingElement( arg[1] ) then
diag := [ arg[1] ];
elif ForAll( arg[1], IsRingElement ) then
diag := arg[1];
fi;
if not IsBound( R ) and IsBound( diag ) and diag <> [ ] and IsHomalgRingElement( diag[1] ) then
R := HomalgRing( diag[1] );
fi;
if not IsBound( diag ) then
return CallFuncList( DiagMat, arg );
elif not IsBound( R ) then
Error( "no homalg ring provided\n" );
fi;
if diag = [ ] then
return HomalgZeroMatrix( 0, 0, R );
fi;
diag := List( diag, a -> HomalgMatrix( [ a ], 1, 1, R ) ); ## a listlist would screw Singular
M := DiagMat( diag );
d := Length( diag );
if nargs > 1 and IsInt( arg[2] ) then
if arg[2] > d then
M := UnionOfRows( M, HomalgZeroMatrix( arg[2] - d, d, R ) );
elif arg[2] < d then
M := CertainRows( M, [ 1 .. arg[2] ] );
fi;
fi;
if nargs > 2 and IsInt( arg[3] ) then
if arg[3] > d then
M := UnionOfColumns( M, HomalgZeroMatrix( NrRows( M ), arg[3] - d, R ) );
elif arg[3] < d then
M := CertainColumns( M, [ 1 .. arg[3] ] );
fi;
fi;
return M;
end );
## <#GAPDoc Label="HomalgScalarMatrix">
## <ManSection>
## <Func Arg="r, n, R" Name="HomalgScalarMatrix" Label="constructor for scalar matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## An immutable unevaluated <M><A>n</A> \times <A>n</A></M> scalar &homalg; matrix over the &homalg; ring <A>R</A> with
## the ring element <A>r</A> as diagonal scalar.
## <Example><![CDATA[
## gap> ZZ := HomalgRingOfIntegers( );
## Z
## gap> d := HomalgScalarMatrix( 2, 3, ZZ );
## <An unevaluated scalar 3 x 3 matrix over an internal ring>
## gap> Display( d );
## [ [ 2, 0, 0 ],
## [ 0, 2, 0 ],
## [ 0, 0, 2 ] ]
## gap> d;
## <A scalar 3 x 3 matrix over an internal ring>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgScalarMatrix,
function( arg ) ## the scalar matrix
local nargs, r, n, R, diag, d, M;
nargs := Length( arg );
if nargs = 0 then
Error( "no arguments provided\n" );
fi;
if IsRingElement( arg[1] ) then
r := arg[1];
else
Error( "the first argument must be a ring element\n" );
fi;
if IsInt( arg[2] ) and arg[2] >= 0 then
n := arg[2];
else
Error( "the second argument must be a non-negative integer\n" );
fi;
if IsHomalgRing( arg[nargs] ) then
R := arg[nargs];
fi;
if not IsBound( R ) then
if IsHomalgRingElement( r ) then
R := HomalgRing( r );
else
Error( "no homalg ring provided\n" );
fi;
fi;
if n = 0 then
return HomalgZeroMatrix( 0, 0, R );
fi;
diag := ListWithIdenticalEntries( n, HomalgMatrix( [ r ], 1, 1, R ) ); ## a listlist would screw Singular
M := DiagMat( diag );
SetIsScalarMatrix( M, true );
return M;
end );
## <#GAPDoc Label="\*:MatrixBaseChange">
## <ManSection>
## <Oper Arg="R, mat" Name="\*" Label="copy a matrix over a different ring"/>
## <Oper Arg="mat, R" Name="\*" Label="copy a matrix over a different ring (right)"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## An immutable evaluated &homalg; matrix over the &homalg; ring <A>R</A> having the
## same entries as the matrix <A>mat</A>. Syntax: <A>R</A> <C>*</C> <A>mat</A> or <A>mat</A> <C>*</C> <A>R</A>
## <Example><![CDATA[
## gap> ZZ := HomalgRingOfIntegers( );
## Z
## gap> Z4 := ZZ / 4;
## Z/( 4 )
## gap> Display( Z4 );
## <A residue class ring>
## gap> d := HomalgDiagonalMatrix( [ 2 .. 4 ], ZZ );
## <An unevaluated diagonal 3 x 3 matrix over an internal ring>
## gap> d2 := Z4 * d; ## or d2 := d * Z4;
## <A 3 x 3 matrix over a residue class ring>
## gap> Display( d2 );
## [ [ 2, 0, 0 ],
## [ 0, 3, 0 ],
## [ 0, 0, 4 ] ]
##
## modulo [ 4 ]
## gap> d;
## <A diagonal 3 x 3 matrix over an internal ring>
## gap> ZeroRows( d );
## [ ]
## gap> ZeroRows( d2 );
## [ 3 ]
## gap> d;
## <A non-zero diagonal 3 x 3 matrix over an internal ring>
## gap> d2;
## <A non-zero 3 x 3 matrix over a residue class ring>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( \*,
"for homalg matrices",
[ IsHomalgRing, IsHomalgMatrix ],
function( R, m )
local RP, mat;
RP := homalgTable( R );
if IsIdenticalObj( HomalgRing( m ), R ) then ## make a copy over the same ring
mat := ShallowCopy( m );
if not IsIdenticalObj( m, mat ) then
return mat;
fi;
fi;
mat := ConvertHomalgMatrix( m, R );
BlindlyCopyMatrixProperties( m, mat );
return mat;
end );
##
InstallMethod( \*,
"for homalg matrices",
[ IsHomalgRing, IsHomalgMatrix and IsZero ], 10001,
function( R, m )
return HomalgZeroMatrix( NrRows( m ), NrColumns( m ), R );
end );
##
InstallMethod( \*,
"for homalg matrices",
[ IsHomalgRing, IsHomalgMatrix and IsOne ], 10001,
function( R, m )
return HomalgIdentityMatrix( NrRows( m ), R );
end );
##
InstallMethod( \*,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgRing ],
function( M, R )
return R * M;
end );
##
InstallMethod( RingMap,
"for homalg rings",
[ IsHomalgMatrix, IsHomalgRing, IsHomalgRing ],
function( images, S, T )
if not ( NrRows( images ) = 1 or NrColumns( images ) = 1 ) then
Error( "the matrix must either has one row or one column\n" );
fi;
return RingMap( EntriesOfHomalgMatrix( images ), S, T );
end );
##
InstallMethod( RingMap,
"constructor for homalg ring maps",
[ IsHomalgMatrix, IsHomalgRing ],
function( mat, R )
local r, indets;
r := NrRows( mat );
if NrColumns( mat ) <> r then
Error( "the matrix is not quadratic\n" );
fi;
indets := Indeterminates( R );
if Length( indets ) <> r then
Error( "the number of indeterminates does not match the number of rows of the matrix\n" );
fi;
indets := HomalgMatrix( indets, r, 1, R );
if not IsIdenticalObj( R, HomalgRing( mat ) ) then
mat := R * mat;
fi;
return RingMap( mat * indets, R, R );
end );
##
InstallMethod( RingMap,
"constructor for homalg ring maps",
[ IsHomalgMatrix ],
function( mat )
return RingMap( mat, HomalgRing( mat ) );
end );
##
InstallGlobalFunction( ListToListList,
function( L, r, c )
local M, i;
M := [ ];
for i in [ 1 .. r ] do
Append( M, [ L{[ (i-1)*c+1 .. i*c ]} ] );
od;
return M;
end );
####################################
#
# View, Print, and Display methods:
#
####################################
##
InstallMethod( ViewObj,
"for weak pointer containers of identity matrices",
[ IsContainerForWeakPointersOnIdentityMatricesRep ],
function( o )
Print( "<A container for weak pointers on identity matrices: counter = ", o!.counter, ", cache_hits = ", o!.cache_hits, ">" );
end );
##
InstallMethod( ViewObj,
"for interal matrix hulls",
[ IsInternalMatrixHull ],
function( o )
Print( "<A hull for a homalg internal matrix>" );
end );
##
InstallMethod( ViewObj,
"for homalg matrices",
[ IsHomalgMatrix ],
function( o )
local R, first_attribute, not_row_or_column_matrix;
R := HomalgRing( o );
first_attribute := true;
if HasIsVoidMatrix( o ) and IsVoidMatrix( o ) then
Print( "<A void" );
elif HasIsInitialMatrix( o ) and IsInitialMatrix( o ) then
Print( "<An initial" );
elif HasIsInitialIdentityMatrix( o ) and IsInitialIdentityMatrix( o ) then
Print( "<An initial identity" );
elif not HasEval( o ) then
Print( "<An unevaluated" );
else
Print( "<A" );
first_attribute := false;
fi;
if not ( HasIsSubidentityMatrix( o ) and IsSubidentityMatrix( o ) )
and HasIsZero( o ) then ## if this method applies and HasIsZero is set we already know that o is a non-zero homalg matrix
Print( " non-zero" );
first_attribute := true;
fi;
not_row_or_column_matrix := not ( ( HasNrRows( o ) and NrRows( o ) = 1 ) or ( HasNrColumns( o ) and NrColumns( o ) = 1 ) );
if not ( HasNrRows( o ) and NrRows( o ) = 1 and HasNrColumns( o ) and NrColumns( o ) = 1 ) then
if HasIsDiagonalMatrix( o ) and IsDiagonalMatrix( o ) then
Print( " diagonal" );
elif HasIsUpperStairCaseMatrix( o ) and IsUpperStairCaseMatrix( o ) and not_row_or_column_matrix then
if not first_attribute then
Print( "n upper staircase" );
else
Print( " upper staircase" );
fi;
elif HasIsStrictUpperTriangularMatrix( o ) and IsStrictUpperTriangularMatrix( o ) then
Print( " strict upper triangular" );
elif HasIsLowerStairCaseMatrix( o ) and IsLowerStairCaseMatrix( o ) and not_row_or_column_matrix then
Print( " lower staircase" );
elif HasIsStrictLowerTriangularMatrix( o ) and IsStrictLowerTriangularMatrix( o ) then
Print( " strict lower triangular" );
elif HasIsUpperTriangularMatrix( o ) and IsUpperTriangularMatrix( o ) and not ( HasNrRows( o ) and NrRows( o ) = 1 ) then
if not first_attribute then
Print( "n upper triangular" );
else
Print( " upper triangular" );
fi;
elif HasIsLowerTriangularMatrix( o ) and IsLowerTriangularMatrix( o ) and not ( HasNrColumns( o ) and NrColumns( o ) = 1 ) then
Print( " lower triangular" );
elif HasIsTriangularMatrix( o ) and IsTriangularMatrix( o ) and not_row_or_column_matrix then
Print( " triangular" );
elif not first_attribute then
first_attribute := fail;
fi;
if first_attribute <> fail then
first_attribute := true;
else
first_attribute := false;
fi;
if HasIsInvertibleMatrix( o ) and IsInvertibleMatrix( o ) then
if not first_attribute then
Print( "n invertible" );
else
Print( " invertible" );
fi;
else
if HasIsRightInvertibleMatrix( o ) and IsRightInvertibleMatrix( o ) then
Print( " right invertible" );
elif HasIsLeftRegular( o ) and IsLeftRegular( o ) then
Print( " left regular" );
fi;
if HasIsLeftInvertibleMatrix( o ) and IsLeftInvertibleMatrix( o ) then
Print( " left invertible" );
elif HasIsRightRegular( o ) and IsRightRegular( o ) then
Print( " right regular" );
fi;
fi;
fi;
if HasIsSubidentityMatrix( o ) and IsSubidentityMatrix( o ) then
Print( " sub-identity" );
fi;
if HasNrRows( o ) then
Print( " ", NrRows( o ), " " );
if not HasNrColumns( o ) then
Print( "x ?" );
fi;
fi;
if HasNrColumns( o ) then
if not HasNrRows( o ) then
Print( " ? " );
fi;
Print( "x ", NrColumns( o ) );
fi;
if IsMutable( o ) and HasEval( o ) then
Print( " mutable" );
fi;
Print( " matrix over a" );
if IsBound( R!.description ) then
Print( R!.description );
elif IsHomalgInternalMatrixRep( o ) then
Print( "n internal" );
fi;
Print( " ring>" );
end );
##
InstallMethod( ViewObj,
"for homalg matrices",
[ IsHomalgMatrix and IsPermutationMatrix ],
function( o )
local R;
R := HomalgRing( o );
if HasEval( o ) then
Print( "<A " );
else
Print( "<An unevaluated " );
fi;
if HasNrRows( o ) then
Print( NrRows( o ), " " );
if not HasNrColumns( o ) then
Print( "x ?" );
fi;
fi;
if HasNrColumns( o ) then
if not HasNrRows( o ) then
Print( "? " );
fi;
Print( "x ", NrColumns( o ) );
fi;
Print( " permutation matrix over a" );
if IsBound( R!.description ) then
Print( R!.description );
elif IsHomalgInternalMatrixRep( o ) then
Print( "n internal" );
fi;
Print( " ring>" );
end );
##
InstallMethod( ViewObj,
"for homalg matrices",
[ IsHomalgMatrix and IsOne ],
function( o )
local R;
R := HomalgRing( o );
if HasEval( o ) then
Print( "<A " );
else
Print( "<An unevaluated " );
fi;
if HasNrRows( o ) then
Print( NrRows( o ), " " );
if not HasNrColumns( o ) then
Print( "x ?" );
fi;
fi;
if HasNrColumns( o ) then
if not HasNrRows( o ) then
Print( "? " );
fi;
Print( "x ", NrColumns( o ) );
fi;
Print( " identity matrix over a" );
if IsBound( R!.description ) then
Print( R!.description );
elif IsHomalgInternalMatrixRep( o ) then
Print( "n internal" );
fi;
Print( " ring>" );
end );
##
InstallMethod( ViewObj,
"for homalg matrices",
[ IsHomalgMatrix and IsZero ],
function( o )
local R;
R := HomalgRing( o );
if HasEval( o ) then
Print( "<A " );
else
Print( "<An unevaluated " );
fi;
if HasNrRows( o ) then
Print( NrRows( o ), " " );
if not HasNrColumns( o ) then
Print( "x ?" );
fi;
fi;
if HasNrColumns( o ) then
if not HasNrRows( o ) then
Print( "? " );
fi;
Print( "x ", NrColumns( o ) );
fi;
Print( " zero matrix over a" );
if IsBound( R!.description ) then
Print( R!.description );
elif IsHomalgInternalMatrixRep( o ) then
Print( "n internal" );
fi;
Print( " ring>" );
end );
##
InstallMethod( Display,
"for internal matrix hulls",
[ IsInternalMatrixHull ],
function( o )
Display( o!.matrix );
end );
##
InstallMethod( Display,
"for homalg matrices",
[ IsHomalgInternalMatrixRep ],
function( o )
Display( Eval( o ) );
end );
##
InstallMethod( Display,
"for homalg matrices",
[ IsHomalgMatrix and IsEmptyMatrix ], 10001,
function( o )
Print( "(an empty ", NrRows( o ), " x ", NrColumns( o ), " matrix)\n" );
end );