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class="nocss"> </span><a href="chap5.html#X850AEC9C7C00AFF5">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7D0EAF527F8514E0">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7A8546EA87E3AE67">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X86CEB1FC7C358777">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X86DFDD25824E2F35">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X85B64BD47F0379C5">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F6BAF8E7F2343EA">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8788CB987A8A18A7">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X82B2C4987D6D5BD3">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X867A947682754A9A">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X81CDBBBE878DC1E5">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X85DA10C07E7E5A3D">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X79C124318588F37F">5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7EEFD7887E96714F">5 </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"><var class="Arg">m</var> × <var class="Ar MakeImmuta (Reference 12.6-4)).<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"><var class="Arg">m</var> × <var class="Ar MakeImmutable (Reference 12.6-4)).<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"><var class="Arg">m</var> × <var class="Arg">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 := 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"><var class="Arg">m</var> × <var class="Arg">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 := 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"><var class="Arg">m</var> × <var class="Arg">n</var></span> <strong cla <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"><var class="Arg">m</var> × <var class="Arg">n</va <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 ], <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 ], <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 <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 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 ); <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.html#X864ACCB08094F <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.html#X864ACCB08094F <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.html#X864ACCB08094F <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.html#X8 <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 );</sp <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.h <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.htm <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> --> -------------------- --> maximum size reached --> -------------------- ¤ Dauer der Verarbeitung: 0.33 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28