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<p><a id="X81274D247D293332" name="X81274D247D293332"></a></p>
<div class="ChapSects"><a href="chap4_mj.html#X81274D247D293332">4 <span class="Heading">Homogeneous Matrices</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7912A7EE7DD9C130">4.1 <span class="Heading">Homogeneous Matrices: Category and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X860DABC0806C4064">4.1-1 IsHomalgMatrixOverGradedRingRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7E3D72117F84D517">4.2 <span class="Heading">Homogeneous Matrices: Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7AC02E03868CB664">4.2-1 MatrixOverGradedRing</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7C1C5AA2813F4F73">4.3 <span class="Heading">Homogeneous Matrices: Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8608086E82701204">4.3-1 DegreesOfEntries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X86C3DD8479FDDB36">4.3-2 NonTrivialDegreePerRow</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X801EBD4D7DAB0672">4.3-3 NonTrivialDegreePerColumn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B109D738475FED4">4.3-4 HomogeneousPartOfMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B28E6B97D1A5E09">4.3-5 IsMatrixOverGradedRingWithHomogeneousEntries</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X81897DD8835ACE8C">4.4 <span class="Heading">Homogeneous Matrices: Operations and Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7DC25A5E83DC3E85">4.4-1 UnderlyingNonGradedRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7D0D1E1E784C3BC7">4.4-2 SetMatElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B56EC5E8545C1B6">4.4-3 AddToMatElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X865E51967E6D0AD3">4.4-4 MatElmAsString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8028986083BEC896">4.4-5 MatElm</a></span>
</div></div>
</div>

<h3>4 <span class="Heading">Homogeneous Matrices</span></h3>

<p>The package <strong class="pkg">GradedRingForHomalg</strong> defines the classes of graded rings, ring elements and homogeneous matrices over such rings. These three objects can be used as data structures defined in <strong class="pkg">MatricesForHomalg</strong> on which the <strong class="pkg">homalg</strong> project can rely to do homological computations over graded rings.</p>

<p>The graded rings most prominently can be used with methods known from general <strong class="pkg">homalg</strong> rings. The methods for doing the computations are presented in the appendix (<a href="chapB_mj.html#X78C55DF7875560DD"><span class="RefLink">B</span></a>), since they are not for external use. The new attributes and operations are documented here.</p>

<p>Since the objects inplemented here are representations from objects elsewhere in the <strong class="pkg">homalg</strong> project (i.e. <strong class="pkg">MatricesForHomalg</strong>), we want to stress that there are many other operations in <strong class="pkg">MatricesForHomalg</strong>, which can be used in connection with the ones presented here. A few of them can be found in the examples and the appendix of this documentation.</p>

<p>Operations within <strong class="pkg">MatricesForHomalg</strong> that take matrices as input and produce a matrix as an output produce homogeneous output for homogeneous input in the following cases: the graded ring in question is either a polynomial ring or the exterior algebra residing in <strong class="pkg">Singular</strong>, and the called operation is one of the following listed below:</p>


<ul>
<li><p><code class="code">SyzygiesGeneratorsOfRows</code></p>

</li>
<li><p><code class="code">SyzygiesGeneratorsOfColumns</code></p>

</li>
<li><p><code class="code">ReducedSyzygiesGeneratorsOfRows</code></p>

</li>
<li><p><code class="code">ReducedSyzygiesGeneratorsOfColumns</code></p>

</li>
<li><p><code class="code">BasisOfRowModule</code></p>

</li>
<li><p><code class="code">BasisOfColumnModule</code></p>

</li>
<li><p><code class="code">ReducedBasisOfRowModule</code></p>

</li>
<li><p><code class="code">ReducedBasisOfColumnModule</code></p>

</li>
<li><p><code class="code">DecideZeroRows</code></p>

</li>
<li><p><code class="code">DecideZeroColumns</code></p>

</li>
<li><p><code class="code">LeftDivide</code></p>

</li>
<li><p><code class="code">RightDivide</code></p>

</li>
</ul>
<p>These operation trigger Gröbner bases computations in <strong class="pkg">Singular</strong>, which are always forced to be performed with a tail reduction by <strong class="pkg">homalg</strong>. In particular, the resulting elements of the Gröbner bases have to be homogeneous.</p>

<p><a id="X7912A7EE7DD9C130" name="X7912A7EE7DD9C130"></a></p>

<h4>4.1 <span class="Heading">Homogeneous Matrices: Category and Representations</span></h4>

<p><a id="X860DABC0806C4064" name="X860DABC0806C4064"></a></p>

<h5>4.1-1 IsHomalgMatrixOverGradedRingRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHomalgMatrixOverGradedRingRep</code>( <var class="Arg">A</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Returns: true or false</p>

<p>The representation of <strong class="pkg">homalg</strong> matrices with entries in a <strong class="pkg">homalg</strong> graded ring.</p>

<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="code">IsMatrixOverGradedRing</code>.)</p>


<div class="example"><pre>
DeclareRepresentation( "IsHomalgMatrixOverGradedRingRep",
        IsMatrixOverGradedRing,
        [ ] );
</pre></div>

<p><a id="X7E3D72117F84D517" name="X7E3D72117F84D517"></a></p>

<h4>4.2 <span class="Heading">Homogeneous Matrices: Constructors</span></h4>

<p><a id="X7AC02E03868CB664" name="X7AC02E03868CB664"></a></p>

<h5>4.2-1 MatrixOverGradedRing</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatrixOverGradedRing</code>( <var class="Arg">mat</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a matrix over a graded ring</p>

<p>Creates a matrix for the graded ring <var class="Arg">S</var>, where <var class="Arg">mat</var> is a matrix over <code class="code">UnderlyingNonGradedRing</code>(<var class="Arg">S</var>).</p>

<p><a id="X7C1C5AA2813F4F73" name="X7C1C5AA2813F4F73"></a></p>

<h4>4.3 <span class="Heading">Homogeneous Matrices: Attributes</span></h4>

<p><a id="X8608086E82701204" name="X8608086E82701204"></a></p>

<h5>4.3-1 DegreesOfEntries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreesOfEntries</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a listlist of degrees/multi-degrees</p>

<p>The matrix of degrees of the matrix <var class="Arg">A</var>.</p>

<p><a id="X86C3DD8479FDDB36" name="X86C3DD8479FDDB36"></a></p>

<h5>4.3-2 NonTrivialDegreePerRow</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonTrivialDegreePerRow</code>( <var class="Arg">A</var>[, <var class="Arg">col_degrees</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of degrees/multi-degrees</p>

<p>The list of first nontrivial degree per row of the matrix <var class="Arg">A</var>.</p>

<p><a id="X801EBD4D7DAB0672" name="X801EBD4D7DAB0672"></a></p>

<h5>4.3-3 NonTrivialDegreePerColumn</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonTrivialDegreePerColumn</code>( <var class="Arg">A</var>[, <var class="Arg">row_degrees</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of degrees/multi-degrees</p>

<p>The list of first nontrivial degree per column of the matrix <var class="Arg">A</var>.</p>

<p><a id="X7B109D738475FED4" name="X7B109D738475FED4"></a></p>

<h5>4.3-4 HomogeneousPartOfMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomogeneousPartOfMatrix</code>( <var class="Arg">A</var>, <var class="Arg">degrees</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a homalg matrix over graded ring</p>

<p>The output is the homogeneous part of the matrix <var class="Arg">A</var> with respect to the given degrees <var class="Arg">degrees</var>. See <var class="Arg">HomogeneousPartOfRingElement</var>.</p>

<p><a id="X7B28E6B97D1A5E09" name="X7B28E6B97D1A5E09"></a></p>

<h5>4.3-5 IsMatrixOverGradedRingWithHomogeneousEntries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMatrixOverGradedRingWithHomogeneousEntries</code>( <var class="Arg">A</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>Checks if every entry in a given matrix <var class="Arg">A</var> over a graded ring is homogeneous.</p>

<p><a id="X81897DD8835ACE8C" name="X81897DD8835ACE8C"></a></p>

<h4>4.4 <span class="Heading">Homogeneous Matrices: Operations and Functions</span></h4>

<p><a id="X7DC25A5E83DC3E85" name="X7DC25A5E83DC3E85"></a></p>

<h5>4.4-1 UnderlyingNonGradedRing</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingNonGradedRing</code>( <var class="Arg">mat</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> ring</p>

<p>The nongraded ring underlying <code class="code">HomalgRing</code>(<var class="Arg">mat</var>).</p>

<p><a id="X7D0D1E1E784C3BC7" name="X7D0D1E1E784C3BC7"></a></p>

<h5>4.4-2 SetMatElm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SetMatElm</code>( <var class="Arg">mat</var>, <var class="Arg">i</var>, <var class="Arg">j</var>, <var class="Arg">r</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Changes the entry (<var class="Arg">i,j</var>) of the matrix <var class="Arg">mat</var> to the value <var class="Arg">r</var>. Here <var class="Arg">R</var> is the graded <strong class="pkg">homalg</strong> ring involved in these computations.</p>

<p><a id="X7B56EC5E8545C1B6" name="X7B56EC5E8545C1B6"></a></p>

<h5>4.4-3 AddToMatElm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AddToMatElm</code>( <var class="Arg">mat</var>, <var class="Arg">i</var>, <var class="Arg">j</var>, <var class="Arg">r</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Changes the entry (<var class="Arg">i,j</var>) of the matrix <var class="Arg">mat</var> by adding the value <var class="Arg">r</var> to it. Here <var class="Arg">R</var> is the (graded) <strong class="pkg">homalg</strong> ring involved in these computations.</p>

<p><a id="X865E51967E6D0AD3" name="X865E51967E6D0AD3"></a></p>

<h5>4.4-4 MatElmAsString</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatElmAsString</code>( <var class="Arg">mat</var>, <var class="Arg">i</var>, <var class="Arg">j</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a string</p>

<p>Returns the entry (<var class="Arg">i,j</var>) of the matrix <var class="Arg">mat</var> as a string. Here <var class="Arg">R</var> is the (graded) <strong class="pkg">homalg</strong> ring involved in these computations.</p>

<p><a id="X8028986083BEC896" name="X8028986083BEC896"></a></p>

<h5>4.4-5 MatElm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatElm</code>( <var class="Arg">mat</var>, <var class="Arg">i</var>, <var class="Arg">j</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a graded ring element</p>

<p>Returns the entry (<var class="Arg">i,j</var>) of the matrix <var class="Arg">mat</var>. Here <var class="Arg">R</var> is the (graded) <strong class="pkg">homalg</strong> ring involved in these computations.</p>


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