Quelle HomalgMatrix.gi
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# SPDX-License-Identifier: GPL-2.0-or-later
# MatricesForHomalg: Matrices for the homalg project
#
# Implementations
#
####################################
#
# 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,
NumberColumns );
####################################
#
# 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 HasNumberRows( S ) then
SetNumberRows( T, NumberRows( S ) );
fi;
if HasNumberColumns( S ) then
SetNumberColumns( T, NumberColumns( 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 ),
NumberRows, NumberRows( M ),
NumberColumns, NumberColumns( M ),
], R );
MatchPropertiesAndAttributes( M, MM,
LIMAT.intrinsic_properties,
LIMAT.intrinsic_attributes,
LIMAT.intrinsic_components,
LIMAT.intrinsic_attributes_do_not_check_their_equality
);
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 ),
NumberRows, NumberRows( M ),
NumberColumns, NumberColumns( 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 ),
NumberRows, NumberRows( M ),
NumberColumns, NumberColumns( M ),
], R );
if not IsIdenticalObj( Eval( M ), Eval( MM ) ) then
MatchPropertiesAndAttributes( M, MM,
LIMAT.intrinsic_properties,
LIMAT.intrinsic_attributes,
LIMAT.intrinsic_components,
LIMAT.intrinsic_attributes_do_not_check_their_equality
);
return MM;
fi;
fi;
if not IsInternalMatrixHull( Eval( M ) ) then
TryNextMethod( );
fi;
return HomalgMatrix( One( R ) * Eval( M )!.matrix, NumberRows( M ), NumberColumns( 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 ),
NumberRows, NumberRows( M ),
NumberColumns, NumberColumns( 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, NumberRows( M ), NumberColumns( M ), R );
SetIsMutableMatrix( MM, true );
return MM;
end );
##
InstallMethod( ShallowCopy,
"for homalg matrices",
[ IsHomalgMatrix and IsOne ],
function( M )
return HomalgIdentityMatrix( NumberRows( 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( NumberRows( M ), HomalgRing( M ) );
end );
##
InstallMethod( ShallowCopy,
"for homalg matrices",
[ IsHomalgMatrix and IsZero ],
function( M )
return HomalgZeroMatrix( NumberRows( M ), NumberColumns( 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( NumberRows( M ), NumberColumns( 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, IsHomalgIntern alRingRep ],
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( 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 );
if not CompareVersionNumbers( GAPInfo.Version, "4.10" ) then
## copied from gap-4.10.2/lib/matobj.gi
# Install fallback methods for m[i,j] which delegate MatElm resp. SetMatElm,
# for old MatrixObj implementation which don't provide them. We lower the rank
# so that these are only used as a last resort.
InstallMethod( \[\], "for a matrix object and two positions",
[ IsMatrixObj, IsPosInt, IsPosInt ],
-RankFilter(IsMatrixObj),
function( m, row, col )
return MatElm( m, row, col );
end );
InstallMethod( \[\]\:\=, "for a matrix object, two positions, and an object",
[ IsMatrixObj and IsMutable, IsPosInt, IsPosInt, IsObject ],
-RankFilter(IsMatrixObj),
function( m, row, col, obj )
SetMatElm( m, row, col, obj );
end );
fi;
##
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 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 := NumberColumns( M );
s := List( [ 1 .. NumberRows( 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 := NumberColumns( M );
s := List( [ 1 .. NumberRows( 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 := NumberColumns( M );
s := [ ];
for i in [ 1 .. NumberRows( M ) ] do
for j in [ 1 .. c ] do
e := 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 := NumberRows( M );
c := NumberColumns( 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 .. NumberColumns( M ) ];
return List( [ 1 .. NumberRows( M ) ], r -> List( cols, c -> M[ r, c ] ) );
end );
##
InstallMethod( EntriesOfHomalgMatrixAsListList,
"for homalg matrices",
[ IsHomalgMatrix and IsHomalgInternalMatrixRep ],
function( M )
local mat;
if IsEmptyMatrix( M ) then
TryNextMethod( );
fi;
mat := Eval( M );
if not IshomalgInternalMatrixHullRep( mat ) then
TryNextMethod( );
fi;
mat := mat!.matrix;
if not IsList( mat ) then
TryNextMethod( );
fi;
return mat;
end );
##
InstallMethod( EntriesOfHomalgMatrix,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return Flat( EntriesOfHomalgMatrixAsListList( M ) );
end );
##
InstallMethod( EntriesOfHomalgRowVector,
"for a homalg row vector",
[ IsHomalgMatrix ],
function( M )
Assert( 0, NumberRows( M ) = 1 );
return EntriesOfHomalgMatrix( M );
end );
##
InstallMethod( EntriesOfHomalgColumnVector,
"for a homalg column vector",
[ IsHomalgMatrix ],
function( M )
Assert( 0, NumberColumns( M ) = 1 );
return EntriesOfHomalgMatrix( 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 .. NumberColumns( 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 HasNumberRows( M1 ) or HasNumberRows( M2 ) then ## trigger as few as possible operations
return IsIdenticalObj( HomalgRing( M1 ), HomalgRing( M2 ) )
and NumberRows( M1 ) = NumberRows( M2 ) and NumberColumns( M1 ) = NumberColumns( M2 );
else ## no other choice
return IsIdenticalObj( HomalgRing( M1 ), HomalgRing( M2 ) )
and NumberColumns( M1 ) = NumberColumns( M2 ) and NumberRows( M1 ) = NumberRows( 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,
LIMAT.intrinsic_attributes_do_not_check_their_equality
);
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,
LIMAT.intrinsic_attributes_do_not_check_their_equality
);
fi;
return true;
fi;
return false;
end );
##
InstallMethod( ZeroMutable,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
return HomalgZeroMatrix( NumberRows( M ), NumberColumns( 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>. If the underlying ring is commutative, the twist is the identity.<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,
NumberRows, NumberColumns( M ),
NumberColumns, NumberRows( M ),
], HomalgRing( M ) );
SetItsInvolution( M, C );
SetItsInvolution( C, M );
return C;
end );
##
InstallMethod( Involution,
"for homalg matrices",
[ IsHomalgMatrix and HasItsInvolution ],
function( M )
return ItsInvolution( M );
end );
## <#GAPDoc Label="TransposedMatrix">
## <ManSection>
## <Meth Arg="M" Name="TransposedMatrix" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## The transpose of the &homalg; matrix <A>M</A>.<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with TransposedMatrix"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( TransposedMatrix,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local C;
C := HomalgMatrixWithAttributes( [
EvalTransposedMatrix, M,
NumberRows, NumberColumns( M ),
NumberColumns, NumberRows( M ),
], HomalgRing( M ) );
SetItsTransposedMatrix( M, C );
SetItsTransposedMatrix( C, M );
return C;
end );
##
InstallMethod( TransposedMatrix,
"for homalg matrices",
[ IsHomalgMatrix and HasItsTransposedMatrix ],
function( M )
return ItsTransposedMatrix( 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 )
plist := ShallowCopy( plist );
ConvertToRangeRep( plist );
return HomalgMatrixWithAttributes( [
EvalCertainRows, [ M, plist ],
NumberRows, Length( plist ),
NumberColumns, NumberColumns( M )
], HomalgRing( M ) );
end );
##
InstallOtherMethod( \[\],
"for a homalg matrix and a positive integer",
[ IsHomalgMatrix, IsInt ],
function( M, r )
return CertainRows( M, [ r ] );
end );
##
InstallOtherMethod( \{\},
"for a homalg matrix and a positive integer",
[ IsHomalgMatrix, IsList ],
CertainRows );
## <#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>[j]</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 )
plist := ShallowCopy( plist );
ConvertToRangeRep( plist );
return HomalgMatrixWithAttributes( [
EvalCertainColumns, [ M, plist ],
NumberColumns, Length( plist ),
NumberRows, NumberRows( M )
], HomalgRing( M ) );
end );
## <#GAPDoc Label="UnionOfRows">
## <ManSection>
## <Func Arg="[R, nr_cols, ]L" Name="UnionOfRows" Label="for a homalg ring, an integer and a list of homalg matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Stack the &homalg; matrices in the list <A>L</A>. The entries of <A>L</A> must be matrices over the &homalg; ring <A>R</A> with <A>nr_cols</A> columns.
## If <A>L</A> is non-empty, <A>R</A> and <A>nr_cols</A> can be omitted.<P/>
##
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with UnionOfRows"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( UnionOfRows,
function( arg )
local R, nr_cols, list;
if IsEmpty( arg ) then
Error( "<arg> must be nonempty" );
elif IsHomalgMatrix( arg[1] ) then
# UnionOfRows( mat1, mat2, ... )
list := arg;
R := HomalgRing( list[1] );
nr_cols := NumberColumns( list[1] );
elif Length( arg ) = 1 and IsList( arg[1] ) then
# UnionOfRows( [ mat1, mat2, ... ] )
if IsEmpty( arg[1] ) then
Error( "<arg>[1] must be nonempty" );
fi;
list := arg[1];
R := HomalgRing( list[1] );
nr_cols := NumberColumns( list[1] );
elif Length( arg ) = 3 and IsHomalgRing( arg[1] ) and IsInt( arg[2] ) and IsList( arg[3] ) then
# UnionOfRows( ring, nr_cols, [ mat1, mat2, ... ] )
R := arg[1];
nr_cols := arg[2];
list := arg[3];
else
Error("usage: UnionOfRows( mat1, mat2, ... ) or UnionOfRows( [ mat1, mat2, ... ] ) or UnionOfRows( ring, nr_cols, [ mat1, mat2, ... ] )");
fi;
return UnionOfRowsOp( R, nr_cols, list );
end );
##
InstallMethod( UnionOfRowsOp,
"of a homalg ring, an integer and a list of homalg matrices",
[ IsHomalgRing, IsInt, IsList ],
function( R, nr_cols, L )
local result;
result := HomalgMatrixWithAttributes( [
EvalUnionOfRows, L,
NumberRows, Sum( List( L, NumberRows ) ),
NumberColumns, nr_cols
], R );
if IsBound( HOMALG_MATRICES.UnionOfRowsEager ) and HOMALG_MATRICES.UnionOfRowsEager = true then
Eval( result );
fi;
return result;
end );
##
InstallGlobalFunction( UnionOfRowsEager,
function( arg )
local nargs;
nargs := Length( arg );
if nargs = 0 then
Error( "<arg> must be nonempty" );
elif Length( arg ) = 1 and IsList( arg[1] ) then
if IsEmpty( arg[1] ) then
Error( "<arg>[1] must be nonempty" );
fi;
arg := arg[1];
fi;
return UnionOfRowsEagerOp( arg, arg[1] );
end );
##
InstallMethod( UnionOfRowsEagerOp,
"of a list of homalg matrices and a homalg matrix",
[ IsList, IsHomalgMatrix ],
function( L, M )
local result;
result := HomalgMatrixWithAttributes( [
EvalUnionOfRows, L,
NumberRows, Sum( List( L, NumberRows ) ),
NumberColumns, NumberColumns( L[1] )
], HomalgRing( L[1] ) );
Eval( result );
return result;
end );
## <#GAPDoc Label="UnionOfColumns">
## <ManSection>
## <Func Arg="[R, nr_rows, ]L" Name="UnionOfColumns" Label="for a homalg ring, an integer and a list of homalg matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Augment the &homalg; matrices in the list <A>L</A>. The entries of <A>L</A> must be matrices over the &homalg; ring <A>R</A> with <A>nr_rows</A> rows.
## If <A>L</A> is non-empty, <A>R</A> and <A>nr_rows</A> can be omitted.<P/>
##
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with UnionOfColumns"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( UnionOfColumns,
function( arg )
local R, nr_rows, list;
if IsEmpty( arg ) then
Error( "<arg> must be nonempty" );
elif IsHomalgMatrix( arg[1] ) then
# UnionOfColumns( mat1, mat2, ... )
list := arg;
R := HomalgRing( list[1] );
nr_rows := NumberRows( list[1] );
elif Length( arg ) = 1 and IsList( arg[1] ) then
# UnionOfColumns( [ mat1, mat2, ... ] )
if IsEmpty( arg[1] ) then
Error( "<arg>[1] must be nonempty" );
fi;
list := arg[1];
R := HomalgRing( list[1] );
nr_rows := NumberRows( list[1] );
elif Length( arg ) = 3 and IsHomalgRing( arg[1] ) and IsInt( arg[2] ) and IsList( arg[3] ) then
# UnionOfColumns( ring, nr_rows, [ mat1, mat2, ... ] )
R := arg[1];
nr_rows := arg[2];
list := arg[3];
else
Error("usage: UnionOfColumns( mat1, mat2, ... ) or UnionOfColumns( [ mat1, mat2, ... ] ) or UnionOfColumns( ring, nr_rows, [ mat1, mat2, ... ] )");
fi;
return UnionOfColumnsOp( R, nr_rows, list );
end );
##
InstallMethod( UnionOfColumnsOp,
"of a list of homalg matrices and a homalg matrix",
[ IsHomalgRing, IsInt, IsList ],
function( R, nr_rows, L )
local result;
result := HomalgMatrixWithAttributes( [
EvalUnionOfColumns, L,
NumberRows, nr_rows,
NumberColumns, Sum( List( L, NumberColumns ) )
], R );
if IsBound( HOMALG_MATRICES.UnionOfColumnsEager ) and HOMALG_MATRICES.UnionOfColumnsEager = true then
Eval( result );
fi;
return result;
end );
##
InstallGlobalFunction( UnionOfColumnsEager,
function( arg )
local nargs;
nargs := Length( arg );
if nargs = 0 then
Error( "<arg> must be nonempty" );
elif Length( arg ) = 1 and IsList( arg[1] ) then
if IsEmpty( arg[1] ) then
Error( "<arg>[1] must be nonempty" );
fi;
arg := arg[1];
fi;
return UnionOfColumnsEagerOp( arg, arg[1] );
end );
##
InstallMethod( UnionOfColumnsEagerOp,
"of a list of homalg matrices and a homalg matrix",
[ IsList, IsHomalgMatrix ],
function( L, M )
local result;
result := HomalgMatrixWithAttributes( [
EvalUnionOfColumns, L,
NumberRows, NumberRows( L[1] ),
NumberColumns, Sum( List( L, NumberColumns ) )
], HomalgRing( L[1] ) );
Eval( result );
return result;
end );
## <#GAPDoc Label="ConvertRowToMatrix">
## <ManSection>
## <Meth Arg="M, r, c" Name="ConvertRowToMatrix" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Fold the row <A>M</A> to an <A>r</A>x<A>c</A>-matrix.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( ConvertRowToMatrix,
"for a homalg matrix and two integers",
[ IsHomalgMatrix, IsInt, IsInt ],
function( M, r, c )
local R, C;
if NumberRows( M ) <> 1 then
Error( "expecting a single row matrix as a first argument\n" );
fi;
if NumberColumns( M ) <> r * c then
Error( "the row has not the expected length\n" );
fi;
R := HomalgRing( M );
if r = 1 then
return M;
elif r * c = 0 or ( HasIsZero( M ) and IsZero( M ) ) then
return HomalgZeroMatrix( r, c, R );
fi;
C := HomalgMatrixWithAttributes( [
EvalConvertRowToMatrix, [ M, r, c ],
NumberRows, r,
NumberColumns, c,
], R );
return C;
end );
## <#GAPDoc Label="ConvertColumnToMatrix">
## <ManSection>
## <Meth Arg="M, r, c" Name="ConvertColumnToMatrix" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Fold the column <A>M</A> to an <A>r</A>x<A>c</A>-matrix.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( ConvertColumnToMatrix,
"for a homalg matrix and two integers",
[ IsHomalgMatrix, IsInt, IsInt ],
function( M, r, c )
local R, C;
if NumberColumns( M ) <> 1 then
Error( "expecting a single column matrix as a first argument\n" );
fi;
if NumberRows( M ) <> r * c then
Error( "the column has not the expected height\n" );
fi;
R := HomalgRing( M );
if c = 1 then
return M;
elif r * c = 0 or ( HasIsZero( M ) and IsZero( M ) ) then
return HomalgZeroMatrix( r, c, R );
fi;
C := HomalgMatrixWithAttributes( [
EvalConvertColumnToMatrix, [ M, r, c ],
NumberRows, r,
NumberColumns, c,
], R );
return C;
end );
## <#GAPDoc Label="ConvertMatrixToRow">
## <ManSection>
## <Meth Arg="M" Name="ConvertMatrixToRow" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Unfold the matrix <A>M</A> row-wise into a row.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( ConvertMatrixToRow,
"for a homalg matrix",
[ IsHomalgMatrix ],
function( M )
local r, c, R, C;
r := NumberRows( M );
if r = 1 then
return M;
fi;
c := NumberColumns( M );
R := HomalgRing( M );
if r = 0 or ( HasIsZero( M ) and IsZero( M ) ) then
return HomalgZeroMatrix( 1, r * c, R );
fi;
C := HomalgMatrixWithAttributes( [
EvalConvertMatrixToRow, M,
NumberRows, 1,
NumberColumns, r * c,
], R );
return C;
end );
## <#GAPDoc Label="ConvertMatrixToColumn">
## <ManSection>
## <Meth Arg="M" Name="ConvertMatrixToColumn" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Unfold the matrix <A>M</A> column-wise into a column.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( ConvertMatrixToColumn,
"for a homalg matrix",
[ IsHomalgMatrix ],
function( M )
local c, r, R, C;
c := NumberColumns( M );
if c = 1 then
return M;
fi;
r := NumberRows( M );
R := HomalgRing( M );
if c = 0 or ( HasIsZero( M ) and IsZero( M ) ) then
return HomalgZeroMatrix( r * c, 1, R );
fi;
C := HomalgMatrixWithAttributes( [
EvalConvertMatrixToColumn, M,
NumberRows, r * c,
NumberColumns, 1,
], R );
return C;
end );
## <#GAPDoc Label="DiagMat">
## <ManSection>
## <Meth Arg="[R, ]list" Name="DiagMat" Label="for a homalg ring and a list of homalg matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Build the block diagonal matrix out of the &homalg; matrices listed in <A>list</A>.
## If <A>list</A> is non-empty, <A>R</A> can be omitted.<P/>
##
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with DiagMat"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( DiagMat,
"of a homalg ring and a list of homalg matrices",
[ IsHomalgRing, IsHomogeneousList ],
function( R, list )
return HomalgMatrixWithAttributes( [
EvalDiagMat, list,
NumberRows, Sum( List( list, NumberRows ) ),
NumberColumns, Sum( List( list, NumberColumns ) )
], R );
end );
##
# convenience
InstallOtherMethod( DiagMat,
"of a list of homalg matrices",
[ IsHomogeneousList ],
function( list )
if IsEmpty( list ) then
Error( "the given list of diagonal blocks is empty\n" );
fi;
return DiagMat( HomalgRing( list[1] ), list );
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 ],
NumberRows, NumberRows( A ) * NumberRows( B ),
NumberColumns, NumberColumns( A ) * NumberColumns ( B )
], HomalgRing( A ) );
end );
## <#GAPDoc Label="DualKroneckerMat">
## <ManSection>
## <Meth Arg="A, B" Name="DualKroneckerMat" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## The dual Kronecker 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 DualKroneckerMat"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( DualKroneckerMat,
"of two homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( A, B )
return HomalgMatrixWithAttributes( [
EvalDualKroneckerMat, [ A, B ],
NumberRows, NumberRows( A ) * NumberRows( B ),
NumberColumns, NumberColumns( A ) * NumberColumns ( 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 ],
NumberRows, NumberRows( A ),
NumberColumns, NumberColumns( 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 ],
NumberRows, NumberRows( A ),
NumberColumns, NumberColumns( 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 ],
NumberRows, NumberRows( A ),
NumberColumns, NumberColumns( 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 ],
NumberRows, NumberRows( A ),
NumberColumns, NumberColumns( 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 ],
NumberRows, NumberRows( A ),
NumberColumns, NumberColumns( B )
], HomalgRing( A ) );
end );
##
InstallMethod( \^,
"for homalg maps",
[ IsHomalgMatrix, IsInt ],
function( A, pow )
local R;
if NumberRows( A ) <> NumberColumns( 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( NumberRows( 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 .. NumberRows( 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 .. NumberColumns( 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 := NumberRows( C );
if not m = NumberColumns( C ) then
Error( "the input ", m, "x", NumberColumns( 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,
NumberRows, NumberColumns( M ),
NumberColumns, NumberRows( 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,
NumberColumns, NumberRows( M ),
NumberRows, NumberColumns( 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( NumberRows( M ), NumberColumns( M ) );
return List( [ 1 .. m ], a -> 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 := NumberRows( M );
c := NumberColumns( 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] ) ) );
if l = [ ] then
return [ Zero( R ) ];
fi;
return l;
end );
##
InstallMethod( MaximalMinors,
"of homalg matrices",
[ IsHomalgMatrix ],
function( M )
return Minors( Minimum( NumberRows( M ), NumberColumns( 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 );
SetNumberRows( mat, 1 ); ## should be obsolete
SetNumberColumns( 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 := NumberRows( M );
c := NumberColumns( 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 := NumberRows( M );
c := NumberColumns( 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 IsList( s ) then
return HomalgMatrix( ListToListList( s, r, c ), r, c, R );
else
Error( "the evaluated string is not in {IsMatrix, IsList}\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="llist, m, n, R" Name="HomalgMatrix" Label="constructor for matrices using a listlist with given dimensions"/>
## <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 IsBound( R!.pre_matrix_constructor ) then
M := R!.pre_matrix_constructor( M ); ## [ 0, x ]
fi;
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 );
elif IsList( M ) and Length( M ) > 0 and IsBound( R!.pre_matrix_constructor ) then
M := R!.pre_matrix_constructor( M ); ## [ 0, x ]
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,
NumberRows, Length( M ),
NumberColumns, Length( M[1] ),
Eval, homalgInternalMatrixHull( M ) );
elif IsList( M ) then
## Objectify:
ObjectifyWithAttributes(
matrix, type,
Eval, homalgInternalMatrixHull( M ) );
if M = [ ] then
SetNumberRows( matrix, 0 );
SetNumberColumns( matrix, 0 );
elif M[1] = [] then
SetNumberRows( matrix, Length( M ) );
SetNumberColumns( 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,
NumberRows, arg[2],
NumberColumns, arg[3],
Eval, M );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
Eval, M );
if nr_rows then
SetNumberRows( matrix, arg[2] );
fi;
if nr_columns then
SetNumberColumns( 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,
NumberRows, arg[2],
NumberColumns, arg[3],
Eval, M );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
Eval, M );
if nr_rows then
SetNumberRows( matrix, arg[2] );
fi;
if nr_columns then
SetNumberColumns( matrix, arg[3] );
fi;
fi;
fi;
return matrix;
end );
## <#GAPDoc Label="HomalgMatrixListList">
## <ManSection>
## <Func Arg="llist, m, n, R" Name="HomalgMatrixListList" Label="constructor for matrices using a listlist with given dimensions"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Special case of <Ref Meth="HomalgMatrix" Label="constructor for matrices using a listlist with given dimensions"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgMatrixListList,
function( llist, m, n, R )
if m > 0 and n > 0 then
Assert( 1, IsList( llist ) and Length( llist ) = m );
Assert( 1, ForAll( llist, list -> IsList( list ) and Length( list ) = n ) );
fi;
return HomalgMatrix( llist, m, n, R );
end );
## <#GAPDoc Label="HomalgRowVector">
## <ManSection>
## <Func Arg="entries, nr_cols, R" Name="HomalgRowVector" Label="constructor for matrices with a single row"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Special case of <Ref Meth="HomalgMatrix" Label="constructor for matrices using a list"/> for matrices with a single row.
## <A>entries</A> must be a list of ring elements.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgRowVector,
function( entries, c, R )
return HomalgMatrixListList( [ entries ], 1, c, R );
end );
## <#GAPDoc Label="HomalgColumnVector">
## <ManSection>
## <Func Arg="entries, nr_rows, R" Name="HomalgColumnVector" Label="constructor for matrices with a single column"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Special case of <Ref Meth="HomalgMatrix" Label="constructor for matrices using a list"/> for matrices with a single column.
## <A>entries</A> must be a list of ring elements.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgColumnVector,
function( entries, r, R )
return HomalgMatrixListList( List( entries, x -> [ x ] ), r, 1, R );
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 )
);
if HOMALG_MATRICES.Eager = true and not IsEmptyMatrix( matrix ) then
Eval( matrix );
fi;
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,
NumberRows, arg[1],
NumberColumns, arg[2],
IsZero, true );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
IsZero, true );
if nr_rows then
SetNumberRows( matrix, arg[1] );
fi;
if nr_columns then
SetNumberColumns( 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,
NumberRows, arg[1],
NumberColumns, 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> n[ 1, 1 ] := "1";;
## gap> 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,
NumberRows, arg[1],
NumberColumns, arg[2],
IsInitialMatrix, true );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
IsInitialMatrix, true );
if nr_rows then
SetNumberRows( matrix, arg[1] );
fi;
if nr_columns then
SetNumberColumns( 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> e[ 1, 2 ] := "1";;
## gap> 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,
NumberRows, arg[1],
NumberColumns, 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,
NumberRows, arg[1],
NumberColumns, arg[2],
IsVoidMatrix, true );
else
## Objectify:
ObjectifyWithAttributes(
matrix, type,
IsVoidMatrix, true );
if nr_rows then
SetNumberRows( matrix, arg[1] );
fi;
if nr_columns then
SetNumberColumns( 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( NumberRows( 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 );
##
InstallGlobalFunction( StringToHomalgColumnMatrix,
function( str, R )
if str in [ "", " ", "[]", "[ ]" ] then
return HomalgZeroMatrix( 0, 1, R );
fi;
return HomalgMatrix( str, Length( SplitString( str, "," ) ), 1, R );
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( NumberRows( m ), NumberColumns( m ), R );
end );
##
InstallMethod( \*,
"for homalg matrices",
[ IsHomalgRing, IsHomalgMatrix and IsOne ], 10001,
function( R, m )
return HomalgIdentityMatrix( NumberRows( m ), R );
end );
##
InstallMethod( \*,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgRing ],
function( M, R )
return R * M;
end );
##
# convenience
InstallMethod( CoercedMatrix,
"for a homalg ring and a homalg matrix",
[ IsHomalgRing, IsHomalgMatrix ],
function( new_ring, M )
return CoercedMatrix( HomalgRing( M ), new_ring, M );
end );
## <#GAPDoc Label="CoercedMatrix">
## <ManSection>
## <Oper Arg="ring_from, ring_to, mat" Name="CoercedMatrix" Label="copy a matrix over a different ring"/>
## <Oper Arg="ring_to, mat" Name="CoercedMatrix" Label="copy a matrix over a different ring (convenience)"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## A copy of the &homalg; matrix <A>mat</A> with &homalg; ring <A>ring_from</A> in the &homalg; ring <A>ring_to</A>.<P/>
## (for the installed standard method see <Ref Meth="Eval" Label="for matrices created with CoercedMatrix"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( CoercedMatrix,
"for two homalg rings and a homalg matrix",
[ IsHomalgRing, IsHomalgRing, IsHomalgMatrix ],
function( old_ring, new_ring, M )
local C;
if not IsIdenticalObj( old_ring, HomalgRing( M ) ) then
Error( "the matrix <M> must be a matrix over the ring <old_ring>" );
fi;
C := HomalgMatrixWithAttributes( [
EvalCoercedMatrix, M
], new_ring );
BlindlyCopyMatrixProperties( M, C );
return C;
end );
##
InstallGlobalFunction( ListToListList,
function( L, r, c )
return List( [ 1 .. r ], i -> L{[ (i-1)*c+1 .. i*c ]} );
end );
## <#GAPDoc Label="CoefficientsWithGivenMonomials">
## <ManSection>
## <Meth Arg="M, monomials" Name="CoefficientsWithGivenMonomials" Label="for two homalg matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Let <C>R := HomalgRing(</C><A>M</A><C>)</C>. <C>monomials</C> must be a &homalg; matrix with the same number
## of columns as <C>M</C> consisting of monomials of <C>R</C>.
## This method computes a &homalg; matrix <C>coeffs</C> (with entries in the coefficients ring of <C>R</C>,
## yet still considered as elements of <C>R</C>) such that <C>M = coeffs * monomials</C>.
## If no such matrix exists, the behavior is undefined.
## If the first argument is a &homalg; ring element, it is viewed as a &homalg; matrix with a single entry.
## If the second argument is a list of monomials, it is viewed as a column matrix with the list elements as entries.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( CoefficientsWithGivenMonomials,
"for two homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( M, monomials )
local R;
if NumberColumns( monomials ) <> NumberColumns( M ) then
Error( "the given matrices must have the same number of columns" );
fi;
R := HomalgRing( M );
return HomalgMatrixWithAttributes( [
EvalCoefficientsWithGivenMonomials, [ M, monomials ],
NumberRows, NumberRows( M ),
NumberColumns, NumberRows( monomials )
], R );
end );
##
InstallMethod( CoefficientsWithGivenMonomials,
"for a homalg column matrix and a list of monomials",
[ IsHomalgMatrix, IsList ],
function( M, monomials )
return CoefficientsWithGivenMonomials( M, HomalgMatrix( monomials, Length( monomials ), 1, HomalgRing( M ) ) );
end );
##
InstallMethod( CoefficientsWithGivenMonomials,
"for a homalg ring element and a homalg column matrix",
[ IsHomalgRingElement, IsHomalgMatrix ],
function( poly, monomials )
return CoefficientsWithGivenMonomials( HomalgMatrix( [ poly ], 1, 1, HomalgRing( poly ) ), monomials );
end );
##
InstallMethod( CoefficientsWithGivenMonomials,
"for a homalg ring element and a list of monomials",
[ IsHomalgRingElement, IsList ],
function( poly, monomials )
return CoefficientsWithGivenMonomials( HomalgMatrix( [ poly ], 1, 1, HomalgRing( poly ) ), monomials );
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( ViewString,
"for interal matrix hulls",
[ IsInternalMatrixHull ],
function( o )
return "<A hull for a homalg internal matrix>";
end );
##
InstallMethod( ViewString,
"for homalg matrices",
[ IsHomalgMatrix ],
function( o )
local R, first_attribute, str, not_row_or_column_matrix;
R := HomalgRing( o );
first_attribute := true;
str :="";
if IsVoidMatrix( o ) then
Append( str, "<A void" );
elif IsInitialMatrix( o ) then
Append( str, "<An initial" );
elif IsInitialIdentityMatrix( o ) then
Append( str, "<An initial identity" );
elif not HasEval( o ) then
Append( str, "<An unevaluated" );
else
Append( str, "<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
Append( str, " non-zero" );
first_attribute := true;
fi;
not_row_or_column_matrix := not ( ( HasNumberRows( o ) and NumberRows( o ) = 1 ) or ( HasNumberColumns( o ) and NumberColumns( o ) = 1 ) );
if not ( HasNumberRows( o ) and NumberRows( o ) = 1 and HasNumberColumns( o ) and NumberColumns( o ) = 1 ) then
if HasIsDiagonalMatrix( o ) and IsDiagonalMatrix( o ) then
Append( str, " diagonal" );
elif HasIsUpperStairCaseMatrix( o ) and IsUpperStairCaseMatrix( o ) and not_row_or_column_matrix then
if not first_attribute then
Append( str, "n upper staircase" );
else
Append( str, " upper staircase" );
fi;
elif HasIsStrictUpperTriangularMatrix( o ) and IsStrictUpperTriangularMatrix( o ) then
Append( str, " strict upper triangular" );
elif HasIsLowerStairCaseMatrix( o ) and IsLowerStairCaseMatrix( o ) and not_row_or_column_matrix then
Append( str, " lower staircase" );
elif HasIsStrictLowerTriangularMatrix( o ) and IsStrictLowerTriangularMatrix( o ) then
Append( str, " strict lower triangular" );
elif HasIsUpperTriangularMatrix( o ) and IsUpperTriangularMatrix( o ) and not ( HasNumberRows( o ) and NumberRows( o ) = 1 ) then
if not first_attribute then
Append( str, "n upper triangular" );
else
Append( str, " upper triangular" );
fi;
elif HasIsLowerTriangularMatrix( o ) and IsLowerTriangularMatrix( o ) and not ( HasNumberColumns( o ) and NumberColumns( o ) = 1 ) then
Append( str, " lower triangular" );
elif HasIsTriangularMatrix( o ) and IsTriangularMatrix( o ) and not_row_or_column_matrix then
Append( str, " 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
Append( str, "n invertible" );
else
Append( str, " invertible" );
fi;
else
if HasIsRightInvertibleMatrix( o ) and IsRightInvertibleMatrix( o ) then
Append( str, " right invertible" );
elif HasIsLeftRegular( o ) and IsLeftRegular( o ) then
Append( str, " left regular" );
fi;
if HasIsLeftInvertibleMatrix( o ) and IsLeftInvertibleMatrix( o ) then
Append( str, " left invertible" );
elif HasIsRightRegular( o ) and IsRightRegular( o ) then
Append( str, " right regular" );
fi;
fi;
fi;
if HasIsSubidentityMatrix( o ) and IsSubidentityMatrix( o ) then
Append( str, " sub-identity" );
fi;
if HasNumberRows( o ) then
Append( str, Concatenation( " ", String( NumberRows( o ) ), " " ) );
if not HasNumberColumns( o ) then
Append( str, "x ?" );
fi;
fi;
if HasNumberColumns( o ) then
if not HasNumberRows( o ) then
Append( str, " ? " );
fi;
Append( str, Concatenation( "x ", String( NumberColumns( o ) ) ) );
fi;
if IsMutable( o ) and HasEval( o ) then
Append( str, " mutable" );
fi;
Append( str, " matrix over a" );
if IsBound( R!.description ) then
Append( str, R!.description );
elif IsHomalgInternalMatrixRep( o ) then
Append( str, "n internal" );
fi;
Append( str, " ring>" );
return str;
end );
##
InstallMethod( ViewString,
"for homalg matrices",
[ IsHomalgMatrix and IsPermutationMatrix ],
function( o )
local str, R;
str := "";
if HasEval( o ) then
Append( str, "<A " );
else
Append( str, "<An unevaluated " );
fi;
if HasNumberRows( o ) then
Append( str, Concatenation( String( NumberRows( o ) ), " " ) );
if not HasNumberColumns( o ) then
Append( str, "x ?" );
fi;
fi;
if HasNumberColumns( o ) then
if not HasNumberRows( o ) then
Append( str, "? " );
fi;
Append( str, Concatenation( "x ", String( NumberColumns( o ) ) ) );
fi;
Append( str, " permutation matrix over a" );
R := HomalgRing( o );
if IsBound( R!.description ) then
Append( str, R!.description );
elif IsHomalgInternalMatrixRep( o ) then
Append( str, "n internal" );
fi;
Append( str, " ring>" );
return str;
end );
##
InstallMethod( ViewString,
"for homalg matrices",
[ IsHomalgMatrix and IsOne ],
function( o )
local str, R;
str := "";
if HasEval( o ) then
Append( str, "<A " );
else
Append( str, "<An unevaluated " );
fi;
if HasNumberRows( o ) then
Append( str, Concatenation( String( NumberRows( o ) ), " " ) );
if not HasNumberColumns( o ) then
Append( str, "x ?" );
fi;
fi;
if HasNumberColumns( o ) then
if not HasNumberRows( o ) then
Append( str, "? " );
fi;
Append( str, Concatenation( "x ", String( NumberColumns( o ) ) ) );
fi;
Append( str, " identity matrix over a" );
R := HomalgRing( o );
if IsBound( R!.description ) then
Append( str, R!.description );
elif IsHomalgInternalMatrixRep( o ) then
Append( str, "n internal" );
fi;
Append( str, " ring>" );
return str;
end );
##
InstallMethod( ViewString,
"for homalg matrices",
[ IsHomalgMatrix and IsZero ],
function( o )
local str, R;
str := "";
if HasEval( o ) then
Append( str, "<A " );
else
Append( str, "<An unevaluated " );
fi;
if HasNumberRows( o ) then
Append( str, Concatenation( String( NumberRows( o ) ), " " ) );
if not HasNumberColumns( o ) then
Append( str, "x ?" );
fi;
fi;
if HasNumberColumns( o ) then
if not HasNumberRows( o ) then
Append( str, "? " );
fi;
Append( str, Concatenation( "x ", String( NumberColumns( o ) ) ) );
fi;
Append( str, " zero matrix over a" );
R := HomalgRing( o );
if IsBound( R!.description ) then
Append( str, R!.description );
elif IsHomalgInternalMatrixRep( o ) then
Append( str, "n internal" );
fi;
Append( str, " ring>" );
return str;
end );
##
InstallMethod( String,
"for homalg matrices",
[ IsHomalgMatrix ],
ViewString );
##
InstallMethod( LaTeXOutput,
"for homalg matrices",
[ IsHomalgMatrix ],
function( m )
local r, c, l, i, j, e;
r := NumberRows( m );
c := NumberColumns( m );
if IsEmptyMatrix( m ) then
return Concatenation( "()_{", String( r ), " \\times ", String( c ), "}" );
fi;
m := EntriesOfHomalgMatrixAsListList( m );
l := "\\left( \\begin{array}";
Append( l, Concatenation( "{", ListWithIdenticalEntries( c, 'r' ), "}\n" ) );
for i in [ 1 .. r ] do
for j in [ 1 .. c ] do
Append( l, " " );
e := m[i][j];
if IsZero( e ) then
Append( l, "\\cdot" );
else
Append( l, LaTeXOutput( m[i][j] ) );
fi;
if j < c then
Append( l, " &" );
fi;
od;
if i < r then
Append( l, " \\\\" );
fi;
Append( l, " \n" );
od;
Append( l, "\\end{array} \\right)" );
return l;
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 ", NumberRows( o ), " x ", NumberColumns( o ), " matrix)\n" );
end );
##
InstallMethod( Trace ,
"for homalg matrices",
[ IsHomalgMatrix ],
function( C )
local R;
if NumberRows( C ) <> NumberColumns( C ) then
Error( "the matrix is not a square matrix\n" );
fi;
R := HomalgRing( C );
if IsZero( C ) then
return Zero( R );
elif IsOne( C ) then
return NumberRows( C ) * One( R );
fi;
return Sum( [ 1 .. NumberRows( C ) ], i -> C[ i, i ] );
end );
[zur Elbe Produktseite wechseln0.82QuellennavigatorsAnalyse erneut starten2026-05-06]
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2026-05-26
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