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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X85DA10C07E7E5A3 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X79C124318588F37 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7EEFD7887E96714 </div></div> </div> <h3>5 <span class="Heading">Matrices</span></h3> <p><a id="X78C552687FF14479" name="X78C552687FF14479"></a></p> <h4>5.1 <span class="Heading">Matrices: Category and Representations</span></h4> <p><a id="X7B68E1057F5F011F" name="X7B68E1057F5F011F"></a></p> <h5>5.1-1 IsHomalgMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>The <strong class="pkg">GAP</strong> category of <strong class="pkg">homalg</strong> matrices.< <div class="example"><pre> DeclareCategory( "IsHomalgMatrix", IsMatrixObj and IsAttributeStoringRep ); </pre></div> <p><a id="X7FE94FC47F460E35" name="X7FE94FC47F460E35"></a></p> <h5>5.1-2 IsHomalgInternalMatrixRep</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>The internal representation of <strong class="pkg">homalg</strong> matrices.</p> <p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsH <p><a id="X7977387186436CDF" name="X7977387186436CDF"></a></p> <h4>5.2 <span class="Heading">Matrices: Constructors</span></h4> <p><a id="X86D290B084AC6638" name="X86D290B084AC6638"></a></p> <h5>5.2-1 HomalgInitialMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>A mutable unevaluated initial <span class="SimpleMath">\(\textit{m} \times \textit{n}\)</ <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">z := HomalgInitialMatrix( 2, 3, zz );< <An initial 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">HasIsZero( z );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">IsZero( z );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">z;</span> <A 2 x 3 mutable matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">HasIsZero( z );</span> false </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">n := HomalgInitialMatrix( 2, 3, zz );< <An initial 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">n[ 1, 1 ] := "1";;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">n[ 2, 3 ] := "1";;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">MakeImmutable( n );</span> <A 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( n );</span> [ [ 1, 0, 0 ], [ 0, 0, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsZero( n );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">n;</span> <A non-zero 2 x 3 matrix over an internal ring> </pre></div> <p><a id="X7CB77009868D369A" name="X7CB77009868D369A"></a></p> <h5>5.2-2 HomalgInitialIdentityMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>A mutable unevaluated initial <span class="SimpleMath">\(\textit{m} \times \textit{m}\)</ <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">id := HomalgInitialIdentityMatrix <An initial identity 3 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">HasIsOne( id );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">IsOne( id );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">id;</span> <A 3 x 3 mutable matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">HasIsOne( id );</span> false </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">e := HomalgInitialIdentityMatrix( <An initial identity 3 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">e[ 1, 2 ] := "1";;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">e[ 2, 1 ] := "-1";;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">MakeImmutable( e );</span> <A 3 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( e );</span> [ [ 1, 1, 0 ], [ -1, 1, 0 ], [ 0, 0, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsOne( e );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">e;</span> <A 3 x 3 matrix over an internal ring> </pre></div> <p><a id="X8309EB7B86953A23" name="X8309EB7B86953A23"></a></p> <h5>5.2-3 HomalgZeroMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>An immutable unevaluated <span class="SimpleMath">\(\textit{m} \times \textit{n}\)</span> < <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">z := HomalgZeroMatrix( 2, 3, zz );</sp <An unevaluated 2 x 3 zero matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( z );</span> [ [ 0, 0, 0 ], [ 0, 0, 0 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">z;</span> <A 2 x 3 zero matrix over an internal ring> </pre></div> <p><a id="X83266B9D7BE740D8" name="X83266B9D7BE740D8"></a></p> <h5>5.2-4 HomalgIdentityMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>An immutable unevaluated <span class="SimpleMath">\(\textit{m} \times \textit{m}\)</span> < <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">id := HomalgIdentityMatrix( 3, zz );< <An unevaluated 3 x 3 identity matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( id );</span> [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">id;</span> <A 3 x 3 identity matrix over an internal ring> </pre></div> <p><a id="X7D2E3472879E28AB" name="X7D2E3472879E28AB"></a></p> <h5>5.2-5 HomalgVoidMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>A void <span class="SimpleMath">\(\textit{m} \times \textit{n}\)</span> <strong class="pkg"> <p><a id="X864ACCB08094F0B7" name="X864ACCB08094F0B7"></a></p> <h5>5.2-6 HomalgMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>An immutable evaluated <span class="SimpleMath">\(\textit{m} \times \textit{n}\)</span> <st <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">m := HomalgMatrix( [ [ 1, 2, 3 ], [ 4, 5, 6 <A 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span> [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">m := HomalgMatrix( [ [ 1, 2, 3 ], [ 4, 5, 6 <A 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span> [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">m := HomalgMatrix( [ 1, 2, 3, 4, 5, 6 ], 2 <A 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span> [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">m := HomalgMatrix( "[ [ 1, 2, 3 ], [ 4, 5, <A 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span> [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">m := HomalgMatrix( "[ [ 1, 2, 3 ], [ 4, 5, <A 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span> [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] </pre></div> <p>It is nevertheless recommended to use the following form to create <strong class="pkg">homalg< <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">m := HomalgMatrix( "[ 1, 2, 3, 4, 5, 6 ]" <A 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span> [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] </pre></div> <p>One can split the input string over several lines using the backslash character '\' to end each <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">m := HomalgMatrix( "[ \</span> <span class="GAPprompt">></span> <span class="GAPinput">1, 2, 3, \</span> <span class="GAPprompt">></span> <span class="GAPinput">4, 5, 6 \</span> <span class="GAPprompt">></span> <span class="GAPinput">]", 2, 3, zz );</span> <A 2 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span> [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] </pre></div> <p><a id="X8246E1D17F96DAE7" name="X8246E1D17F96DAE7"></a></p> <h5>5.2-7 HomalgMatrixListList</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Special case of <code class="func">HomalgMatrix</code> (<a href="chap5_mj.html#X864ACCB080 <p><a id="X7B127B5584CD012D" name="X7B127B5584CD012D"></a></p> <h5>5.2-8 HomalgRowVector</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Special case of <code class="func">HomalgMatrix</code> (<a href="chap5_mj.html#X864ACCB080 <p><a id="X871AF271843FF2B5" name="X871AF271843FF2B5"></a></p> <h5>5.2-9 HomalgColumnVector</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Special case of <code class="func">HomalgMatrix</code> (<a href="chap5_mj.html#X864ACCB080 <p><a id="X872D39C678D0C4AE" name="X872D39C678D0C4AE"></a></p> <h5>5.2-10 HomalgDiagonalMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>An immutable unevaluated diagonal <strong class="pkg">homalg</strong> matrix over the <strong <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">d := HomalgDiagonalMatrix( [ 1, 2, 3 ] <An unevaluated diagonal 3 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( d );</span> [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 3 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">d;</span> <A diagonal 3 x 3 matrix over an internal ring> </pre></div> <p><a id="X81225377833C4644" name="X81225377833C4644"></a></p> <h5><code>5.2-11 \*</code></h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \*< <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \*< <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>An immutable evaluated <strong class="pkg">homalg</strong> matrix over the <strong class="pkg <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">Z4 := zz / 4;</span> Z/( 4 ) <span class="GAPprompt">gap></span> <span class="GAPinput">Display( Z4 );</span> <A residue class ring> <span class="GAPprompt">gap></span> <span class="GAPinput">d := HomalgDiagonalMatrix( [ 2 .. 4 ], <An unevaluated diagonal 3 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">d2 := Z4 * d; ## or d2 := d * Z4;</span> <A 3 x 3 matrix over a residue class ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( d2 );</span> [ [ 2, 0, 0 ], [ 0, 3, 0 ], [ 0, 0, 4 ] ] modulo [ 4 ] <span class="GAPprompt">gap></span> <span class="GAPinput">d;</span> <A diagonal 3 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">ZeroRows( d );</span> [ ] <span class="GAPprompt">gap></span> <span class="GAPinput">ZeroRows( d2 );</span> [ 3 ] <span class="GAPprompt">gap></span> <span class="GAPinput">d;</span> <A non-zero diagonal 3 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">d2;</span> <A non-zero 3 x 3 matrix over a residue class ring> </pre></div> <p><a id="X7C4E49D287011DCD" name="X7C4E49D287011DCD"></a></p> <h5>5.2-12 CoercedMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>A copy of the <strong class="pkg">homalg</strong> matrix <var class="Arg">mat</var> with <strong cl <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7D92ECFC8030CF40" name="X7D92ECFC8030CF40"></a></p> <h4>5.3 <span class="Heading">Matrices: Properties</span></h4> <p><a id="X858B5AF57D5BC90A" name="X858B5AF57D5BC90A"></a></p> <h5>5.3-1 IsZero</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if the <strong class="pkg">homalg</strong> matrix <var class="Arg">A</var> is a zero matrix, <p>(for the installed standard method see <code class="func">IsZeroMatrix</code> (<a href="chapB <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">A := HomalgMatrix( "[ 2 ]", zz );</span> <A 1 x 1 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Z2 := zz / 2;</span> Z/( 2 ) <span class="GAPprompt">gap></span> <span class="GAPinput">A := Z2 * A;</span> <A 1 x 1 matrix over a residue class ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( A );</span> [ [ 2 ] ] modulo [ 2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsZero( A );</span> true </pre></div> <p><a id="X814D78347858EC13" name="X814D78347858EC13"></a></p> <h5>5.3-2 IsOne</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if the <strong class="pkg">homalg</strong> matrix <var class="Arg">A</var> is an identity ma <p>(for the installed standard method see <code class="func">IsIdentityMatrix</code> (<a href="c <p><a id="X7813653578F174AB" name="X7813653578F174AB"></a></p> <h5>5.3-3 IsUnitFree</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X8612CB4A82D6D79E" name="X8612CB4A82D6D79E"></a></p> <h5>5.3-4 IsPermutationMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7EEE3E9780EBA607" name="X7EEE3E9780EBA607"></a></p> <h5>5.3-5 IsSpecialSubidentityMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X8672364D79EBCC5D" name="X8672364D79EBCC5D"></a></p> <h5>5.3-6 IsSubidentityMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7EF95CAD78BDE12F" name="X7EF95CAD78BDE12F"></a></p> <h5>5.3-7 IsLeftRegular</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X87C369D27D6AAF68" name="X87C369D27D6AAF68"></a></p> <h5>5.3-8 IsRightRegular</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X856E1D217A47EE8C" name="X856E1D217A47EE8C"></a></p> <h5>5.3-9 IsInvertibleMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7A4FA27C80BC42D1" name="X7A4FA27C80BC42D1"></a></p> <h5>5.3-10 IsLeftInvertibleMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7E43FDE57E8449B6" name="X7E43FDE57E8449B6"></a></p> <h5>5.3-11 IsRightInvertibleMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7BFC9266823F2C15" name="X7BFC9266823F2C15"></a></p> <h5>5.3-12 IsEmptyMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7EEC8E768178696E" name="X7EEC8E768178696E"></a></p> <h5>5.3-13 IsDiagonalMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p>Check if the <strong class="pkg">homalg</strong> matrix <var class="Arg">A</var> is a diagonal mat <p>(for the installed standard method see <code class="func">IsDiagonalMatrix</code> (<a href="c <p><a id="X7848E6A0783B7428" name="X7848E6A0783B7428"></a></p> <h5>5.3-14 IsScalarMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X8740E71C799C0BCC" name="X8740E71C799C0BCC"></a></p> <h5>5.3-15 IsUpperTriangularMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X853A5B988306DBFE" name="X853A5B988306DBFE"></a></p> <h5>5.3-16 IsLowerTriangularMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7976C42B7FA905EC" name="X7976C42B7FA905EC"></a></p> <h5>5.3-17 IsStrictUpperTriangularMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7B0C78AF8056D650" name="X7B0C78AF8056D650"></a></p> <h5>5.3-18 IsStrictLowerTriangularMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X81A2C3F67C99A3C2" name="X81A2C3F67C99A3C2"></a></p> <h5>5.3-19 IsUpperStairCaseMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7B3A5DE1860373F0" name="X7B3A5DE1860373F0"></a></p> <h5>5.3-20 IsLowerStairCaseMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7BAAE75A8660D7A5" name="X7BAAE75A8660D7A5"></a></p> <h5>5.3-21 IsTriangularMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7F520F89821A8602" name="X7F520F89821A8602"></a></p> <h5>5.3-22 IsBasisOfRowsMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7D46613983DC5302" name="X7D46613983DC5302"></a></p> <h5>5.3-23 IsBasisOfColumnsMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X86445AD281024339" name="X86445AD281024339"></a></p> <h5>5.3-24 IsReducedBasisOfRowsMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7E6BB540865C0344" name="X7E6BB540865C0344"></a></p> <h5>5.3-25 IsReducedBasisOfColumnsMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7E6E51517822CB3F" name="X7E6E51517822CB3F"></a></p> <h5>5.3-26 IsInitialMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X7EE624707ACEC26E" name="X7EE624707ACEC26E"></a></p> <h5>5.3-27 IsInitialIdentityMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X802794217F56DE51" name="X802794217F56DE51"></a></p> <h5>5.3-28 IsVoidMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="code">true</code> or <code class="code">false</code></p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X86F766077C89558F" name="X86F766077C89558F"></a></p> <h4>5.4 <span class="Heading">Matrices: Attributes</span></h4> <p><a id="X7C72971F7D0CA3C8" name="X7C72971F7D0CA3C8"></a></p> <h5>5.4-1 NumberRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Nu <p>Returns: a nonnegative integer</p> <p>The number of rows of the matrix <var class="Arg">A</var>.</p> <p>(for the installed standard method see <code class="func">NumberRows</code> (<a href="chapB_m <p><a id="X847D45BF7F2BC67C" name="X847D45BF7F2BC67C"></a></p> <h5>5.4-2 NumberColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Nu <p>Returns: a nonnegative integer</p> <p>The number of columns of the matrix <var class="Arg">A</var>.</p> <p>(for the installed standard method see <code class="func">NumberColumns</code> (<a href="chap <p><a id="X83045F6F82C180E1" name="X83045F6F82C180E1"></a></p> <h5>5.4-3 DeterminantMat</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ De <p>Returns: a ring element</p> <p>The determinant of the quadratic matrix <var class="Arg">A</var>.</p> <p>You can invoke it with <code class="code">Determinant</code>( <var class="Arg">A</var> ).</p> <p>(for the installed standard method see <code class="func">Determinant</code> (<a href="chapB_ <p><a id="X828225E0857B1FDA" name="X828225E0857B1FDA"></a></p> <h5>5.4-4 ZeroRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ze <p>Returns: a (possibly empty) list of positive integers</p> <p>The list of zero rows of the matrix <var class="Arg">A</var>.</p> <p>(for the installed standard method see <code class="func">ZeroRows</code> (<a href="chapB_mj. <p><a id="X870D761F7AB96D12" name="X870D761F7AB96D12"></a></p> <h5>5.4-5 ZeroColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ze <p>Returns: a (possibly empty) list of positive integers</p> <p>The list of zero columns of the matrix <var class="Arg">A</var>.</p> <p>(for the installed standard method see <code class="func">ZeroColumns</code> (<a href="chapB_ <p><a id="X7991ED337C73065A" name="X7991ED337C73065A"></a></p> <h5>5.4-6 NonZeroRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ No <p>Returns: a (possibly empty) list of positive integers</p> <p>The list of nonzero rows of the matrix <var class="Arg">A</var>.</p> <p><a id="X7F335DCB7B8781E4" name="X7F335DCB7B8781E4"></a></p> <h5>5.4-7 NonZeroColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ No <p>Returns: a (possibly empty) list of positive integers</p> <p>The list of nonzero columns of the matrix <var class="Arg">A</var>.</p> <p><a id="X7B7A073D7E1FAEA4" name="X7B7A073D7E1FAEA4"></a></p> <h5>5.4-8 PositionOfFirstNonZeroEntryPerRow</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Po <p>Returns: a list of nonnegative integers</p> <p>The list of positions of the first nonzero entry per row of the matrix <var class="Arg">A</var>, el <p><a id="X83B389A97A703E42" name="X83B389A97A703E42"></a></p> <h5>5.4-9 PositionOfFirstNonZeroEntryPerColumn</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Po <p>Returns: a list of nonnegative integers</p> <p>The list of positions of the first nonzero entry per column of the matrix <var class="Arg">A</var> <p><a id="X862841E68674FA2A" name="X862841E68674FA2A"></a></p> <h5>5.4-10 RowRankOfMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ro <p>Returns: a nonnegative integer</p> <p>The row rank of the matrix <var class="Arg">A</var>.</p> <p><a id="X7C61862E81CABD51" name="X7C61862E81CABD51"></a></p> <h5>5.4-11 ColumnRankOfMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>Returns: a nonnegative integer</p> <p>The column rank of the matrix <var class="Arg">A</var>.</p> <p><a id="X7EFCE38281AE60F9" name="X7EFCE38281AE60F9"></a></p> <h5>5.4-12 LeftInverse</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>A left inverse <span class="SimpleMath">\(C\)</span> of the matrix <var class="Arg">M</var>. If no <p>(for the installed standard method see <code class="func">LeftInverse</code> (<a href="chap5_ <p><a id="X87614CA48493B63F" name="X87614CA48493B63F"></a></p> <h5>5.4-13 RightInverse</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>A right inverse <span class="SimpleMath">\(C\)</span> of the matrix <var class="Arg">M</var>. If n <p>(for the installed standard method see <code class="func">RightInverse</code> (<a href="chap5 <p><a id="X7809E0507E882674" name="X7809E0507E882674"></a></p> <h5>5.4-14 CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>Returns: a list of integers</p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix (row convention).</p> <p><a id="X7938E13A7EF4ADB1" name="X7938E13A7EF4ADB1"></a></p> <h5>5.4-15 CoefficientsOfNumeratorOfHilbertPoincareSeries</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>Returns: a list of integers</p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix (row convention).</p> <p><a id="X781E2CDB8743B1C6" name="X781E2CDB8743B1C6"></a></p> <h5>5.4-16 UnreducedNumeratorOfHilbertPoincareSeries</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Un <p>Returns: a univariate polynomial with rational coefficients</p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix (row convention).</p> <p><a id="X7C44039382DD5D91" name="X7C44039382DD5D91"></a></p> <h5>5.4-17 NumeratorOfHilbertPoincareSeries</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Nu <p>Returns: a univariate polynomial with rational coefficients</p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix (row convention).</p> <p><a id="X7B93B7D082A50E61" name="X7B93B7D082A50E61"></a></p> <h5>5.4-18 HilbertPoincareSeries</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Hi <p>Returns: a univariate rational function with rational coefficients</p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix (row convention).</p> <p><a id="X84299BAB807A1E13" name="X84299BAB807A1E13"></a></p> <h5>5.4-19 HilbertPolynomial</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Hi <p>Returns: a univariate polynomial with rational coefficients</p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix (row convention).</p> <p><a id="X7BC36CC67CB09858" name="X7BC36CC67CB09858"></a></p> <h5>5.4-20 AffineDimension</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Af <p>Returns: an integer</p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix (row convention).</p> <p><a id="X87C428A079000336" name="X87C428A079000336"></a></p> <h5>5.4-21 AffineDegree</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Af <p>Returns: a nonnegative integer</p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix (row convention).</p> <p><a id="X82A1B55879AB1742" name="X82A1B55879AB1742"></a></p> <h5>5.4-22 ProjectiveDegree</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <p>Returns: a nonnegative integer</p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix (row convention).</p> <p><a id="X791B772A7E368A88" name="X791B772A7E368A88"></a></p> <h5>5.4-23 ConstantTermOfHilbertPolynomialn</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>Returns: an integer</p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix (row convention).</p> <p><a id="X835972A77F02C5BB" name="X835972A77F02C5BB"></a></p> <h5>5.4-24 MatrixOfSymbols</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ma <p>Returns: an integer</p> <p><var class="Arg">A</var> is a <strong class="pkg">homalg</strong> matrix.</p> <p><a id="X80FA5AE87E8591BC" name="X80FA5AE87E8591BC"></a></p> <h4>5.5 <span class="Heading">Matrices: Operations and Functions</span></h4> <p><a id="X81BBF79C79C3B6DF" name="X81BBF79C79C3B6DF"></a></p> <h5>5.5-1 HomalgRing</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ho <p>Returns: a <strong class="pkg">homalg</strong> ring</p> <p>The <strong class="pkg">homalg</strong> ring of the <strong class="pkg">homalg</strong> matrix <v <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">d := HomalgDiagonalMatrix( [ 2 .. 4 ], <An unevaluated diagonal 3 x 3 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgRing( d );</span> Z <span class="GAPprompt">gap></span> <span class="GAPinput">IsIdenticalObj( R, zz );</span> true </pre></div> <p><a id="X7FBAA11B8008D936" name="X7FBAA11B8008D936"></a></p> <h5>5.5-2 LeftInverse</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <p>Returns: a <strong class="pkg">homalg</strong> matrix or fail</p> <p>The left inverse of the matrix <var class="Arg">RI</var>. The lazy version of this operation is <co <div class="example"><pre> InstallMethod( LeftInverse, "for homalg matrices", [ IsHomalgMatrix ], function( RI ) local Id, LI; Id := HomalgIdentityMatrix( NumberColumns( RI ), HomalgRing( RI ) ); LI := RightDivide( Id, RI ); ## ( cf. [BR08, Subsection 3.1.3] ) ## CAUTION: for the following SetXXX RightDivide is assumed ## NOT to be lazy evaluated!!! SetIsLeftInvertibleMatrix( RI, IsHomalgMatrix( LI ) ); if IsBool( LI ) then return fail; fi; if HasIsInvertibleMatrix( RI ) and IsInvertibleMatrix( RI ) then SetIsInvertibleMatrix( LI, true ); else SetIsRightInvertibleMatrix( LI, true ); fi; SetRightInverse( LI, RI ); SetNumberColumns( LI, NumberRows( RI ) ); if NumberRows( RI ) = NumberColumns( RI ) then ## a left inverse of a ring element is unique ## and coincides with the right inverse SetRightInverse( RI, LI ); SetLeftInverse( LI, RI ); fi; return LI; end ); </pre></div> <p><a id="X7AAD17D47839BCAE" name="X7AAD17D47839BCAE"></a></p> <h5>5.5-3 RightInverse</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <p>Returns: a <strong class="pkg">homalg</strong> matrix or fail</p> <p>The right inverse of the matrix <var class="Arg">LI</var>. The lazy version of this operation is <c <div class="example"><pre> InstallMethod( RightInverse, "for homalg matrices", [ IsHomalgMatrix ], function( LI ) local Id, RI; Id := HomalgIdentityMatrix( NumberRows( LI ), HomalgRing( LI ) ); RI := LeftDivide( LI, Id ); ## ( cf. [BR08, Subsection 3.1.3] ) ## CAUTION: for the following SetXXX LeftDivide is assumed ## NOT to be lazy evaluated!!! SetIsRightInvertibleMatrix( LI, IsHomalgMatrix( RI ) ); if IsBool( RI ) then return fail; fi; if HasIsInvertibleMatrix( LI ) and IsInvertibleMatrix( LI ) then SetIsInvertibleMatrix( RI, true ); else SetIsLeftInvertibleMatrix( RI, true ); fi; SetLeftInverse( RI, LI ); SetNumberRows( RI, NumberColumns( LI ) ); if NumberRows( LI ) = NumberColumns( LI ) then ## a right inverse of a ring element is unique ## and coincides with the left inverse SetLeftInverse( LI, RI ); SetRightInverse( RI, LI ); fi; return RI; end ); </pre></div> <p><a id="X7A7E42C179142727" name="X7A7E42C179142727"></a></p> <h5>5.5-4 LeftInverseLazy</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>A lazy evaluated left inverse <span class="SimpleMath">\(C\)</span> of the matrix <var class="A <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7FA3E7617EED7E1E" name="X7FA3E7617EED7E1E"></a></p> <h5>5.5-5 RightInverseLazy</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>A lazy evaluated right inverse <span class="SimpleMath">\(C\)</span> of the matrix <var class=" <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X800FA81F7C42BFEA" name="X800FA81F7C42BFEA"></a></p> <h5>5.5-6 Involution</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ In <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>The twisted transpose of the <strong class="pkg">homalg</strong> matrix <var class="Arg">M</var> <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7D0D35B582D9C0B0" name="X7D0D35B582D9C0B0"></a></p> <h5>5.5-7 TransposedMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tr <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>The transpose of the <strong class="pkg">homalg</strong> matrix <var class="Arg">M</var>.</p> <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7CF5CE79796001F6" name="X7CF5CE79796001F6"></a></p> <h5>5.5-8 CertainRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ce <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>The matrix of which the <span class="SimpleMath">\(i\)</span>-th row is the <span class="Simple <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X8256AF2A840B19C4" name="X8256AF2A840B19C4"></a></p> <h5>5.5-9 CertainColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ce <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>The matrix of which the <span class="SimpleMath">\(j\)</span>-th column is the <span class="Sim <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7D6D0BDF854C9EBC" name="X7D6D0BDF854C9EBC"></a></p> <h5>5.5-10 UnionOfRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Un <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Stack the <strong class="pkg">homalg</strong> matrices in the list <var class="Arg">L</var>. The e <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7FF9661D85EC46B1" name="X7FF9661D85EC46B1"></a></p> <h5>5.5-11 UnionOfColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Un <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Augment the <strong class="pkg">homalg</strong> matrices in the list <var class="Arg">L</var>. Th <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7E61390E79B663E8" name="X7E61390E79B663E8"></a></p> <h5>5.5-12 ConvertRowToMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Fold the row <var class="Arg">M</var> to an <var class="Arg">r</var>x<var class="Arg">c</var>-matrix <p><a id="X853EA6C87EDDF6EF" name="X853EA6C87EDDF6EF"></a></p> <h5>5.5-13 ConvertColumnToMatrix</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Fold the column <var class="Arg">M</var> to an <var class="Arg">r</var>x<var class="Arg">c</var>-mat <p><a id="X7ED5C86379A647F2" name="X7ED5C86379A647F2"></a></p> <h5>5.5-14 ConvertMatrixToRow</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Unfold the matrix <var class="Arg">M</var> row-wise into a row.</p> <p><a id="X84C7C1D07DB6FBAA" name="X84C7C1D07DB6FBAA"></a></p> <h5>5.5-15 ConvertMatrixToColumn</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Unfold the matrix <var class="Arg">M</var> column-wise into a column.</p> <p><a id="X7C7830BE847D84B4" name="X7C7830BE847D84B4"></a></p> <h5>5.5-16 DiagMat</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Di <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Build the block diagonal matrix out of the <strong class="pkg">homalg</strong> matrices listed <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7CDA5D848468A0AA" name="X7CDA5D848468A0AA"></a></p> <h5>5.5-17 KroneckerMat</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Kr <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>The Kronecker (or tensor) product of the two <strong class="pkg">homalg</strong> matrices <var c <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7ECF744B7DE82BED" name="X7ECF744B7DE82BED"></a></p> <h5>5.5-18 DualKroneckerMat</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Du <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>The dual Kronecker product of the two <strong class="pkg">homalg</strong> matrices <var class=" <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7D1A074278B415BE" name="X7D1A074278B415BE"></a></p> <h5><code>5.5-19 \*</code></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</p> <p>The product of the ring element <var class="Arg">a</var> with the <strong class="pkg">homalg</str <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X87C773DA85B21ADF" name="X87C773DA85B21ADF"></a></p> <h5><code>5.5-20 \+</code></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</p> <p>The sum of the two <strong class="pkg">homalg</strong> matrices <var class="Arg">A</var> and <var cl <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X784B57617B24208C" name="X784B57617B24208C"></a></p> <h5><code>5.5-21 \-</code></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</p> <p>The difference of the two <strong class="pkg">homalg</strong> matrices <var class="Arg">A</var> a <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7F5961D78754157B" name="X7F5961D78754157B"></a></p> <h5><code>5.5-22 \*</code></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</p> <p>The matrix product of the two <strong class="pkg">homalg</strong> matrices <var class="Arg">A</v <p>(for the installed standard method see <code class="func">Eval</code> (<a href="chapC_mj.html <p><a id="X7E2074A77AFF518A" name="X7E2074A77AFF518A"></a></p> <h5><code>5.5-23 \=</code></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>Check if the <strong class="pkg">homalg</strong> matrices <var class="Arg">A</var> and <var class= <p>(for the installed standard method see <code class="func">AreEqualMatrices</code> (<a href="c <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</spa Z <span class="GAPprompt">gap></span> <span class="GAPinput">A := HomalgMatrix( "[ 1 ]", zz );</span> <A 1 x 1 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">B := HomalgMatrix( "[ 3 ]", zz );</span> <A 1 x 1 matrix over an internal ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Z2 := zz / 2;</span> Z/( 2 ) <span class="GAPprompt">gap></span> <span class="GAPinput">A := Z2 * A;</span> <A 1 x 1 matrix over a residue class ring> <span class="GAPprompt">gap></span> <span class="GAPinput">B := Z2 * B;</span> <A 1 x 1 matrix over a residue class ring> <span class="GAPprompt">gap></span> <span class="GAPinput">Display( A );</span> [ [ 1 ] ] modulo [ 2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Display( B );</span> [ [ 3 ] ] modulo [ 2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">A = B;</span> true </pre></div> <p><a id="X85887BBB86F0A08B" name="X85887BBB86F0A08B"></a></p> <h5>5.5-24 GetColumnIndependentUnitPositions</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ge <p>Returns: a (possibly empty) list of pairs of positive integers</p> <p>The list of column independet unit position of the matrix <var class="Arg">A</var>. We say that a u <p>(for the installed standard method see <code class="func">GetColumnIndependentUnitPositi <p><a id="X824AB44184DD63B0" name="X824AB44184DD63B0"></a></p> <h5>5.5-25 GetRowIndependentUnitPositions</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ge <p>Returns: a (possibly empty) list of pairs of positive integers</p> <p>The list of row independet unit position of the matrix <var class="Arg">A</var>. We say that a unit < <p>(for the installed standard method see <code class="func">GetRowIndependentUnitPositions< <p><a id="X7A1969A17979FC49" name="X7A1969A17979FC49"></a></p> <h5>5.5-26 GetUnitPosition</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ge <p>Returns: a (possibly empty) list of pairs of positive integers</p> <p>The position <span class="SimpleMath">\([i,j]\)</span> of the first unit <var class="Arg">A</va <p>(for the installed standard method see <code class="func">GetUnitPosition</code> (<a href="ch <p><a id="X781B1C0C80529B09" name="X781B1C0C80529B09"></a></p> <h5>5.5-27 Eliminate</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ El <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Eliminate the independents <var class="Arg">indets</var> from the matrix (or list of ring eleme <p><a id="X80ADBE0D82CC6E85" name="X80ADBE0D82CC6E85"></a></p> <h5>5.5-28 BasisOfRowModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ba <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Let <span class="SimpleMath">\(R\)</span> be the ring over which <var class="Arg">M</var> is defin <p><a id="X868CDA327D6C8DDC" name="X868CDA327D6C8DDC"></a></p> <h5>5.5-29 BasisOfColumnModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ba <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Let <span class="SimpleMath">\(R\)</span> be the ring over which <var class="Arg">M</var> is defin <p><a id="X7F851EC7861170D1" name="X7F851EC7861170D1"></a></p> <h5>5.5-30 DecideZeroRows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ De <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Let <var class="Arg">A</var> and <var class="Arg">B</var> be matrices having the same number of colu <p><a id="X86C97DBB787BAD6D" name="X86C97DBB787BAD6D"></a></p> <h5>5.5-31 DecideZeroColumns</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ De <p>Returns: a <strong class="pkg">homalg</strong> matrix</p> <p>Let <var class="Arg">A</var> and <var class="Arg">B</var> be matrices having the same number of rows |
