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<p><a id="X7B2993CB7B012115" name="X7B2993CB7B012115"></a></p>
<div class="ChapSects"><a href="chapA_mj.html#X7B2993CB7B012115">A <span class="Heading">The Matrix Tool Operations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA_mj.html#X7988F0AF7D87FD23">A.1 <span class="Heading">The Tool Operations <em>without</em> a Fallback Method</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA_mj.html#X7912E42C81296637">A.2 <span class="Heading">The Tool Operations with a Fallback Method</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA_mj.html#X84C78C6B7C0890BF">A.2-1 MonomialMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA_mj.html#X7F9F9C978703E871">A.2-2 RandomMatrixBetweenGradedFreeLeftModules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA_mj.html#X82E813D982331C0B">A.2-3 RandomMatrixBetweenGradedFreeRightModules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA_mj.html#X7EA1DC07822DE322">A.2-4 Diff</a></span>
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<h3>A <span class="Heading">The Matrix Tool Operations</span></h3>

<p>The functions listed below are components of the <code class="code">homalgTable</code> object stored in the ring. They are only indirectly accessible through standard methods that invoke them.</p>

<p><a id="X7988F0AF7D87FD23" name="X7988F0AF7D87FD23"></a></p>

<h4>A.1 <span class="Heading">The Tool Operations <em>without</em> a Fallback Method</span></h4>

<p>There are matrix methods for which <strong class="pkg">homalg</strong> needs a <code class="code">homalgTable</code> entry for non-internal rings, as it cannot provide a suitable fallback. Below is the list of these <code class="code">homalgTable</code> entries.</p>

<p><a id="X7912E42C81296637" name="X7912E42C81296637"></a></p>

<h4>A.2 <span class="Heading">The Tool Operations with a Fallback Method</span></h4>

<p>These are the methods for which it is recommended for performance reasons to have a <code class="code">homalgTable</code> entry for non-internal rings. <strong class="pkg">homalg</strong> only provides a generic fallback method.</p>

<p><a id="X84C78C6B7C0890BF" name="X84C78C6B7C0890BF"></a></p>

<h5>A.2-1 MonomialMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MonomialMatrix</code>( <var class="Arg">d</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>

<p>The column matrix of <var class="Arg">d</var>-th monomials of the <strong class="pkg">homalg</strong> graded ring <var class="Arg">R</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := MonomialMatrix( 2, S );</span>
<A ? x 1 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberRows( m );</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">m;</span>
<A 6 x 1 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span>
x^2,
x*y,
x*z,
y^2,
y*z,
z^2
(over a graded ring) 
</pre></div>

<p><a id="X7F9F9C978703E871" name="X7F9F9C978703E871"></a></p>

<h5>A.2-2 RandomMatrixBetweenGradedFreeLeftModules</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RandomMatrixBetweenGradedFreeLeftModules</code>( <var class="Arg">degreesS</var>, <var class="Arg">degreesT</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>

<p>A random <span class="SimpleMath">\(r \times c \)</span>-matrix between the graded free <em>left</em> modules <span class="SimpleMath">\(\textit{R}^{(-\textit{degreesS})} \to \textit{R}^{(-\textit{degreesT})}\)</span>, where <span class="SimpleMath">\(r = \)</span><code class="code">Length</code><span class="SimpleMath">\((\)</span><var class="Arg">degreesS</var><span class="SimpleMath">\()\)</span> and <span class="SimpleMath">\(c = \)</span><code class="code">Length</code><span class="SimpleMath">\((\)</span><var class="Arg">degreesT</var><span class="SimpleMath">\()\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rand := RandomMatrixBetweenGradedFreeLeftModules( [ 2, 3, 4 ], [ 1, 2 ], S );</span>
<A 3 x 2 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">#Display( rand );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#a-2*b+2*c,                                                2,                 </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#a^2-a*b+b^2-2*b*c+5*c^2,                                  3*c,               </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#2*a^3-3*a^2*b+2*a*b^2+3*a^2*c+a*b*c-2*b^2*c-3*b*c^2-2*c^3,a^2-4*a*b-3*a*c-c^2</span>
</pre></div>

<p><a id="X82E813D982331C0B" name="X82E813D982331C0B"></a></p>

<h5>A.2-3 RandomMatrixBetweenGradedFreeRightModules</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RandomMatrixBetweenGradedFreeRightModules</code>( <var class="Arg">degreesT</var>, <var class="Arg">degreesS</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>

<p>A random <span class="SimpleMath">\(r \times c \)</span>-matrix between the graded free <em>right</em> modules <span class="SimpleMath">\(\textit{R}^{(-\textit{degreesS})} \to \textit{R}^{(-\textit{degreesT})}\)</span>, where <span class="SimpleMath">\(r = \)</span><code class="code">Length</code><span class="SimpleMath">\((\)</span><var class="Arg">degreesT</var><span class="SimpleMath">\()\)</span> and <span class="SimpleMath">\(c = \)</span><code class="code">Length</code><span class="SimpleMath">\((\)</span><var class="Arg">degreesS</var><span class="SimpleMath">\()\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rand := RandomMatrixBetweenGradedFreeRightModules( [ 1, 2 ], [ 2, 3, 4 ], S );</span>
<A 2 x 3 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">#Display( rand );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#a-2*b-c,a*b+b^2-b*c,2*a^3-a*b^2-4*b^3+4*a^2*c-3*a*b*c-b^2*c+a*c^2+5*b*c^2-2*c^3,</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#-5,     -2*a+c,     -2*a^2-a*b-2*b^2-3*a*c                                      </span>
</pre></div>

<p><a id="X7EA1DC07822DE322" name="X7EA1DC07822DE322"></a></p>

<h5>A.2-4 Diff</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Diff</code>( <var class="Arg">D</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>

<p>If <var class="Arg">D</var> is a <span class="SimpleMath">\(f \times p\)</span>-matrix and <var class="Arg">N</var> is a <span class="SimpleMath">\(g \times q\)</span>-matrix then <span class="SimpleMath">\(H=Diff(\)</span><var class="Arg">D</var>,<var class="Arg">N</var><span class="SimpleMath">\()\)</span> is an <span class="SimpleMath">\(fg \times pq\)</span>-matrix whose entry <span class="SimpleMath">\(H[g*(i-1)+j,q*(k-1)+l]\)</span> is the result of differentiating <var class="Arg">N</var><span class="SimpleMath">\([j,l]\)</span> by the differential operator corresponding to <var class="Arg">D</var><span class="SimpleMath">\([i,k]\)</span>. (Here we follow the Macaulay2 convention.)</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c" * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x,2*y,   \</span>
<span class="GAPprompt">></span> <span class="GAPinput">y,a-b^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">z,y-b    \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 3, 2, S );</span>
<A 3 x 2 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x^2-a*y^3,x^3-z^2*y,x*y-b,x*z-c, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x,        x*y,      a-b,  x*a*b  \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 2, 4, S );</span>
<A 2 x 4 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := Diff( D, N );</span>
<A 6 x 8 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( H );</span>
2*x,     3*x^2, y,z,  -6*a*y^2,-2*z^2,2*x,0,  
1,       y,     0,a*b,0,       2*x,   0,  0,  
-3*a*y^2,-z^2,  x,0,  -y^3,    0,     0,  0,  
0,       x,     0,0,  0,       0,     1,  b*x,
0,       -2*y*z,0,x,  -3*a*y^2,-z^2,  x+1,0,  
0,       0,     0,0,  0,       x,     1,  -a*x
</pre></div>


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