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<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (MatricesForHomalg) - Appendix D: The subpackage ResidueClassRingForHomalg as a s <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> <script src="manual.js" type="text/javascript"></script> <script type="text/javascript">overwriteStyle();</script> </head> <body class="chapD" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <p id="mathjaxlink" class="pcenter"><a href="chapD_mj.html">[MathJax on]</a></p> <p><a id="X7F3BA9AE7A0D245D" name="X7F3BA9AE7A0D245D"></a></p> <div 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/><span class="nocss"> </span><a href="chapD.html#X7EFC928C7E59CEAE">D <span class="ContSS"><br /><span class="nocss"> </span><a href="chapD.html#X7E78F7D6796C7016">D </div></div> </div> <h3>D <span class="Heading">The subpackage <strong class="pkg">ResidueClassRingForHomalg</st <p><a id="X84978AF3878A8375" name="X84978AF3878A8375"></a></p> <h4>D.1 <span class="Heading">The Mandatory Basic Operations</span></h4> <p><a id="X7AB980C5791BA204" name="X7AB980C5791BA204"></a></p> <h5>D.1-1 BasisOfRowModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> BasisOfRowModule := function( M ) local Mrel; Mrel := StackedRelations( M ); Mrel := HomalgResidueClassMatrix( BasisOfRowModule( Mrel ), HomalgRing( M ) ); return GetRidOfObsoleteRows( Mrel ); end, </pre></div> <p><a id="X7F2B3332793FACA3" name="X7F2B3332793FACA3"></a></p> <h5>D.1-2 BasisOfColumnModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> BasisOfColumnModule := function( M ) local Mrel; Mrel := AugmentedRelations( M ); Mrel := HomalgResidueClassMatrix( BasisOfColumnModule( Mrel ), HomalgRing( M ) ); return GetRidOfObsoleteColumns( Mrel ); end, </pre></div> <p><a id="X83E072E1790A7D38" name="X83E072E1790A7D38"></a></p> <h5>D.1-3 DecideZeroRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> DecideZeroRows := function( A, B ) local Brel; Brel := StackedRelations( B ); Brel := BasisOfRowModule( Brel ); return HomalgResidueClassMatrix( DecideZeroRows( Eval( A ), Brel ), HomalgRing( A ) ); end, </pre></div> <p><a id="X841426A87A1A20E4" name="X841426A87A1A20E4"></a></p> <h5>D.1-4 DecideZeroColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> DecideZeroColumns := function( A, B ) local Brel; Brel := AugmentedRelations( B ); Brel := BasisOfColumnModule( Brel ); return HomalgResidueClassMatrix( DecideZeroColumns( Eval( A ), Brel ), HomalgRing( A ) ); end, </pre></div> <p><a id="X80F4836F7F175B12" name="X80F4836F7F175B12"></a></p> <h5>D.1-5 SyzygiesGeneratorsOfRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> SyzygiesGeneratorsOfRows := function( M ) local R, ring_rel, rel, S; R := HomalgRing( M ); ring_rel := RingRelations( R ); rel := MatrixOfRelations( ring_rel ); if IsHomalgRingRelationsAsGeneratorsOfRightIdeal( ring_rel ) then rel := Involution( rel ); fi; rel := DiagMat( ListWithIdenticalEntries( NumberColumns( M ), rel ) ); S := SyzygiesGeneratorsOfRows( Eval( M ), rel ); S := HomalgResidueClassMatrix( S, R ); S := GetRidOfObsoleteRows( S ); if IsZero( S ) then SetIsLeftRegular( M, true ); fi; return S; end, </pre></div> <p><a id="X7899768C8304A59E" name="X7899768C8304A59E"></a></p> <h5>D.1-6 SyzygiesGeneratorsOfColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> SyzygiesGeneratorsOfColumns := function( M ) local R, ring_rel, rel, S; R := HomalgRing( M ); ring_rel := RingRelations( R ); rel := MatrixOfRelations( ring_rel ); if IsHomalgRingRelationsAsGeneratorsOfLeftIdeal( ring_rel ) then rel := Involution( rel ); fi; rel := DiagMat( ListWithIdenticalEntries( NumberRows( M ), rel ) ); S := SyzygiesGeneratorsOfColumns( Eval( M ), rel ); S := HomalgResidueClassMatrix( S, R ); S := GetRidOfObsoleteColumns( S ); if IsZero( S ) then SetIsRightRegular( M, true ); fi; return S; end, </pre></div> <p><a id="X78BC2E8E7A78CC82" name="X78BC2E8E7A78CC82"></a></p> <h5>D.1-7 BasisOfRowsCoeff</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> BasisOfRowsCoeff := function( M, T ) local Mrel, TT, bas, nz; Mrel := StackedRelations( M ); TT := HomalgVoidMatrix( HomalgRing( Mrel ) ); bas := BasisOfRowsCoeff( Mrel, TT ); bas := HomalgResidueClassMatrix( bas, HomalgRing( M ) ); nz := NonZeroRows( bas ); SetEval( T, CertainRows( CertainColumns( TT, [ 1 .. NumberRows( M ) ] ), nz ) ); ResetFilterObj( T, IsVoidMatrix ); ## the generic BasisOfRowsCoeff will assume that ## ( NumberRows( B ) = 0 ) = IsZero( B ) return CertainRows( bas, nz ); end, </pre></div> <p><a id="X7D2E9D797877FFBD" name="X7D2E9D797877FFBD"></a></p> <h5>D.1-8 BasisOfColumnsCoeff</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> BasisOfColumnsCoeff := function( M, T ) local Mrel, TT, bas, nz; Mrel := AugmentedRelations( M ); TT := HomalgVoidMatrix( HomalgRing( Mrel ) ); bas := BasisOfColumnsCoeff( Mrel, TT ); bas := HomalgResidueClassMatrix( bas, HomalgRing( M ) ); nz := NonZeroColumns( bas ); SetEval( T, CertainColumns( CertainRows( TT, [ 1 .. NumberColumns( M ) ] ), nz ) ); ResetFilterObj( T, IsVoidMatrix ); ## the generic BasisOfColumnsCoeff will assume that ## ( NumberColumns( B ) = 0 ) = IsZero( B ) return CertainColumns( bas, nz ); end, </pre></div> <p><a id="X7F10DC697D2B828D" name="X7F10DC697D2B828D"></a></p> <h5>D.1-9 DecideZeroRowsEffectively</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> DecideZeroRowsEffectively := function( A, B, T ) local Brel, TT, red; Brel := StackedRelations( B ); TT := HomalgVoidMatrix( HomalgRing( Brel ) ); red := DecideZeroRowsEffectively( Eval( A ), Brel, TT ); SetEval( T, CertainColumns( TT, [ 1 .. NumberRows( B ) ] ) ); ResetFilterObj( T, IsVoidMatrix ); return HomalgResidueClassMatrix( red, HomalgRing( A ) ); end, </pre></div> <p><a id="X7B68B9F27BC02520" name="X7B68B9F27BC02520"></a></p> <h5>D.1-10 DecideZeroColumnsEffectively</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> DecideZeroColumnsEffectively := function( A, B, T ) local Brel, TT, red; Brel := AugmentedRelations( B ); TT := HomalgVoidMatrix( HomalgRing( Brel ) ); red := DecideZeroColumnsEffectively( Eval( A ), Brel, TT ); SetEval( T, CertainRows( TT, [ 1 .. NumberColumns( B ) ] ) ); ResetFilterObj( T, IsVoidMatrix ); return HomalgResidueClassMatrix( red, HomalgRing( A ) ); end, </pre></div> <p><a id="X7DF62C5D7D2E6A6E" name="X7DF62C5D7D2E6A6E"></a></p> <h5>D.1-11 RelativeSyzygiesGeneratorsOfRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> RelativeSyzygiesGeneratorsOfRows := function( M, M2 ) local M2rel, S; M2rel := StackedRelations( M2 ); S := SyzygiesGeneratorsOfRows( Eval( M ), M2rel ); S := HomalgResidueClassMatrix( S, HomalgRing( M ) ); S := GetRidOfObsoleteRows( S ); if IsZero( S ) then SetIsLeftRegular( M, true ); fi; return S; end, </pre></div> <p><a id="X852F9FD8837D97A5" name="X852F9FD8837D97A5"></a></p> <h5>D.1-12 RelativeSyzygiesGeneratorsOfColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <div class="example"><pre> RelativeSyzygiesGeneratorsOfColumns := function( M, M2 ) local M2rel, S; M2rel := AugmentedRelations( M2 ); S := SyzygiesGeneratorsOfColumns( Eval( M ), M2rel ); S := HomalgResidueClassMatrix( S, HomalgRing( M ) ); S := GetRidOfObsoleteColumns( S ); if IsZero( S ) then SetIsRightRegular( M, true ); fi; return S; end, </pre></div> <p><a id="X83E14F457ADC297D" name="X83E14F457ADC297D"></a></p> <h4>D.2 <span class="Heading">The Mandatory Tool Operations</span></h4> <p>Here we list those matrix operations for which <strong class="pkg">homalg</strong> provides no <p><a id="X81CD6BAB7CA73AFC" name="X81CD6BAB7CA73AFC"></a></p> <h5>D.2-1 InitialMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">InitialMatrix</code> (<a href="chapB.html#X7DBA33F083A317B5"><span c <div class="example"><pre> InitialMatrix := C -> HomalgInitialMatrix( NumberRows( C ), NumberColumns( C ), AmbientRing( HomalgRing( C ) ) ), </pre></div> <p><a id="X7D0F99857E280142" name="X7D0F99857E280142"></a></p> <h5>D.2-2 InitialIdentityMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">InitialIdentityMatrix</code> (<a href="chapB.html#X84179BE87E7DCE <div class="example"><pre> InitialIdentityMatrix := C -> HomalgInitialIdentityMatrix( NumberRows( C ), AmbientRing( HomalgRing( C ) ) ), </pre></div> <p><a id="X80393225841391E7" name="X80393225841391E7"></a></p> <h5>D.2-3 ZeroMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">ZeroMatrix</code> (<a href="chapB.html#X785390E38396CAEB"><span clas <div class="example"><pre> ZeroMatrix := C -> HomalgZeroMatrix( NumberRows( C ), NumberColumns( C ), AmbientRing( HomalgRing( C ) ) ), </pre></div> <p><a id="X811B306C81435D87" name="X811B306C81435D87"></a></p> <h5>D.2-4 IdentityMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">IdentityMatrix</code> (<a href="chapB.html#X87BFF3567DEEBEF4"><span <div class="example"><pre> IdentityMatrix := C -> HomalgIdentityMatrix( NumberRows( C ), AmbientRing( HomalgRing( C ) ) ), </pre></div> <p><a id="X7B322C637FC26E2D" name="X7B322C637FC26E2D"></a></p> <h5>D.2-5 Involution</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">Involution</code> (<a href="chapB.html#X85884C3178473521"><span clas <div class="example"><pre> Involution := function( M ) local N, R; N := Involution( Eval( M ) ); R := HomalgRing( N ); if not ( HasIsCommutative( R ) and IsCommutative( R ) and HasIsReducedModuloRingRelations( M ) and IsReducedModuloRingRelations( M ) ) then ## reduce the matrix N w.r.t. the ring relations N := DecideZero( N, HomalgRing( M ) ); fi; return N; end, </pre></div> <p><a id="X816CBEA6790E0C31" name="X816CBEA6790E0C31"></a></p> <h5>D.2-6 TransposedMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">TransposedMatrix</code> (<a href="chapB.html#X7AD2EEE680DF472B"><sp <div class="example"><pre> TransposedMatrix := function( M ) local N, R; N := TransposedMatrix( Eval( M ) ); R := HomalgRing( N ); if not ( HasIsCommutative( R ) and IsCommutative( R ) and HasIsReducedModuloRingRelations( M ) and IsReducedModuloRingRelations( M ) ) then ## reduce the matrix N w.r.t. the ring relations N := DecideZero( N, HomalgRing( M ) ); fi; return N; end, </pre></div> <p><a id="X7C5B29A37B13A53D" name="X7C5B29A37B13A53D"></a></p> <h5>D.2-7 CertainRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">CertainRows</code> (<a href="chapB.html#X7B6FC3267CD9EE9D"><span cla <div class="example"><pre> CertainRows := function( M, plist ) local N; N := CertainRows( Eval( M ), plist ); if not ( HasIsReducedModuloRingRelations( M ) and IsReducedModuloRingRelations( M ) ) then ## reduce the matrix N w.r.t. the ring relations N := DecideZero( N, HomalgRing( M ) ); fi; return N; end, </pre></div> <p><a id="X84CBE51981BA2C77" name="X84CBE51981BA2C77"></a></p> <h5>D.2-8 CertainColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">CertainColumns</code> (<a href="chapB.html#X78EADFC67D17CF04"><span <div class="example"><pre> CertainColumns := function( M, plist ) local N; N := CertainColumns( Eval( M ), plist ); if not ( HasIsReducedModuloRingRelations( M ) and IsReducedModuloRingRelations( M ) ) then ## reduce the matrix N w.r.t. the ring relations N := DecideZero( N, HomalgRing( M ) ); fi; return N; end, </pre></div> <p><a id="X8510678E8569799E" name="X8510678E8569799E"></a></p> <h5>D.2-9 UnionOfRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">UnionOfRows</code> (<a href="chapB.html#X7DEB535782A3323E"><span cla <div class="example"><pre> UnionOfRows := function( L ) local N; N := UnionOfRows( List( L, Eval ) ); if not ForAll( L, HasIsReducedModuloRingRelations and IsReducedModuloRingRelations ) then ## reduce the matrix N w.r.t. the ring relations N := DecideZero( N, HomalgRing( L[1] ) ); fi; return N; end, </pre></div> <p><a id="X862F756A7FC0F0D4" name="X862F756A7FC0F0D4"></a></p> <h5>D.2-10 UnionOfColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">UnionOfColumns</code> (<a href="chapB.html#X7DF5DB55836D13A7"><span <div class="example"><pre> UnionOfColumns := function( L ) local N; N := UnionOfColumns( List( L, Eval ) ); if not ForAll( L, HasIsReducedModuloRingRelations and IsReducedModuloRingRelations ) then ## reduce the matrix N w.r.t. the ring relations N := DecideZero( N, HomalgRing( L[1] ) ); fi; return N; end, </pre></div> <p><a id="X802BEBF5790D4167" name="X802BEBF5790D4167"></a></p> <h5>D.2-11 DiagMat</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">DiagMat</code> (<a href="chapB.html#X86C5B86981FA1F9A"><span class=" <div class="example"><pre> DiagMat := function( e ) local N; N := DiagMat( List( e, Eval ) ); if not ForAll( e, HasIsReducedModuloRingRelations and IsReducedModuloRingRelations ) then ## reduce the matrix N w.r.t. the ring relations N := DecideZero( N, HomalgRing( e[1] ) ); fi; return N; end, </pre></div> <p><a id="X85F1DDDB864AF265" name="X85F1DDDB864AF265"></a></p> <h5>D.2-12 KroneckerMat</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">KroneckerMat</code> (<a href="chapB.html#X82202A6A7FAB7174"><span cl <div class="example"><pre> KroneckerMat := function( A, B ) local N; N := KroneckerMat( Eval( A ), Eval( B ) ); if not ForAll( [ A, B ], HasIsReducedModuloRingRelations and IsReducedModuloRingRelations ) then ## reduce the matrix N w.r.t. the ring relations N := DecideZero( N, HomalgRing( A ) ); fi; return N; end, </pre></div> <p><a id="X7E996A50863FE76C" name="X7E996A50863FE76C"></a></p> <h5>D.2-13 DualKroneckerMat</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">DualKroneckerMat</code> (<a href="chapB.html#X87E0747D7FEEAC76"><sp <div class="example"><pre> DualKroneckerMat := function( A, B ) local N; N := DualKroneckerMat( Eval( A ), Eval( B ) ); if not ForAll( [ A, B ], HasIsReducedModuloRingRelations and IsReducedModuloRingRelations ) then ## reduce the matrix N w.r.t. the ring relations N := DecideZero( N, HomalgRing( A ) ); fi; return N; end, </pre></div> <p><a id="X7A5D229384E9D19C" name="X7A5D229384E9D19C"></a></p> <h5>D.2-14 MulMat</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">MulMat</code> (<a href="chapB.html#X828F8C7785EEC3D1"><span class="R <div class="example"><pre> MulMat := function( a, A ) return DecideZero( EvalRingElement( a ) * Eval( A ), HomalgRing( A ) ); end, MulMatRight := function( A, a ) return DecideZero( Eval( A ) * EvalRingElement( a ), HomalgRing( A ) ); end, </pre></div> <p><a id="X83E1AEC781AE1274" name="X83E1AEC781AE1274"></a></p> <h5>D.2-15 AddMat</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">AddMat</code> (<a href="chapB.html#X7B0B12F080A90039"><span class="R <div class="example"><pre> AddMat := function( A, B ) return DecideZero( Eval( A ) + Eval( B ), HomalgRing( A ) ); end, </pre></div> <p><a id="X79A1C9297BE0C09A" name="X79A1C9297BE0C09A"></a></p> <h5>D.2-16 SubMat</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">SubMat</code> (<a href="chapB.html#X7FE11AA27AE7D2D7"><span class="R <div class="example"><pre> SubMat := function( A, B ) return DecideZero( Eval( A ) - Eval( B ), HomalgRing( A ) ); end, </pre></div> <p><a id="X875447A686949D59" name="X875447A686949D59"></a></p> <h5>D.2-17 Compose</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">Compose</code> (<a href="chapB.html#X7D491D957E63C3A4"><span class=" <div class="example"><pre> Compose := function( A, B ) return DecideZero( Eval( A ) * Eval( B ), HomalgRing( A ) ); end, </pre></div> <p><a id="X78D7BABE806B82FA" name="X78D7BABE806B82FA"></a></p> <h5>D.2-18 IsZeroMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>(--> <code class="func">IsZeroMatrix</code> (<a href="chapB.html#X849BB912798A01EB"><span cl <div class="example"><pre> IsZeroMatrix := M -> IsZero( DecideZero( Eval( M ), HomalgRing( M ) ) ), </pre></div> <p><a id="X85CE26418598FACE" name="X85CE26418598FACE"></a></p> <h5>D.2-19 NumberRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a nonnegative integer</p> <p>(--> <code class="func">NumberRows</code> (<a href="chapB.html#X7F4D7FAF821DA1C2"><span clas <div class="example"><pre> NumberRows := C -> NumberRows( Eval( C ) ), </pre></div> <p><a id="X79A76B1A7CB57518" name="X79A76B1A7CB57518"></a></p> <h5>D.2-20 NumberColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a nonnegative integer</p> <p>(--> <code class="func">NumberColumns</code> (<a href="chapB.html#X7DFA534B7AFA2E17"><span c <div class="example"><pre> NumberColumns := C -> NumberColumns( Eval( C ) ), </pre></div> <p><a id="X80831B287AB565BA" name="X80831B287AB565BA"></a></p> <h5>D.2-21 Determinant</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: an element of ambient <strong class="pkg">homalg</strong> ring</p> <p>(--> <code class="func">Determinant</code> (<a href="chapB.html#X80A573257D7F2E1A"><span cla <div class="example"><pre> Determinant := C -> DecideZero( Determinant( Eval( C ) ), HomalgRing( C ) ), </pre></div> <p><a id="X7A537DB185A0F67C" name="X7A537DB185A0F67C"></a></p> <h4>D.3 <span class="Heading">Some of the Recommended Tool Operations</span></h4> <p>Here we list those matrix operations for which <strong class="pkg">homalg</strong> does provid <p><a id="X848EB509816E8A7D" name="X848EB509816E8A7D"></a></p> <h5>D.3-1 AreEqualMatrices</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>(--> <code class="func">AreEqualMatrices</code> (<a href="chapB.html#X7871FE5478BFC167"><sp <div class="example"><pre> AreEqualMatrices := function( A, B ) return IsZero( DecideZero( Eval( A ) - Eval( B ), HomalgRing( A ) ) ); end, </pre></div> <p><a id="X80122FB3846A6BA5" name="X80122FB3846A6BA5"></a></p> <h5>D.3-2 IsOne</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>(--> <code class="func">IsIdentityMatrix</code> (<a href="chapB.html#X80C1856D82172268"><sp <div class="example"><pre> IsIdentityMatrix := M -> IsOne( DecideZero( Eval( M ), HomalgRing( M ) ) ), </pre></div> <p><a id="X87B8E7137DC97A71" name="X87B8E7137DC97A71"></a></p> <h5>D.3-3 IsDiagonalMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>(--> <code class="func">IsDiagonalMatrix</code> (<a href="chapB.html#X7B6420E88418316B"><sp <div class="example"><pre> IsDiagonalMatrix := M -> IsDiagonalMatrix( DecideZero( Eval( M ), HomalgRing( M ) ) ), </pre></div> <p><a id="X7EFC928C7E59CEAE" name="X7EFC928C7E59CEAE"></a></p> <h5>D.3-4 ZeroRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">ZeroRows</code> (<a href="chapB.html#X872B70367F412945"><span class= <div class="example"><pre> ZeroRows := C -> ZeroRows( DecideZero( Eval( C ), HomalgRing( C ) ) ), </pre></div> <p><a id="X7E78F7D6796C7016" name="X7E78F7D6796C7016"></a></p> <h5>D.3-5 ZeroColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <strong class="pkg">homalg</strong> matrix over the ambient ring</p> <p>(--> <code class="func">ZeroColumns</code> (<a href="chapB.html#X7A469E6D7EA63BB6"><span cla <div class="example"><pre> ZeroColumns := C -> ZeroColumns( DecideZero( Eval( C ), HomalgRing( C ) ) ), </pre></div> <div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA </body> </html>
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