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<title>GAP (LinearAlgebraForCAP) - Chapter 1: Category of Matrices</title>
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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chapInd.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap0.html">[Previous Chapter]</a>    <a href="chap2.html">[Next Chapter]</a>   </div>

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<p><a id="X79A2BF518637FFB7" name="X79A2BF518637FFB7"></a></p>
<div class="ChapSects"><a href="chap1.html#X79A2BF518637FFB7">1 <span class="Heading">Category of Matrices</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X86EC0F0A78ECBC10">1.1 <span class="Heading">Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X79D9727280354457">1.1-1 MatrixCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7C6DB0457A656406">1.1-2 VectorSpaceMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7EB162887B8F8708">1.1-3 VectorSpaceObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X818AE01E7C9D21A5">1.1-4 MatrixCategoryObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7A0CAF7F7CE754E7">1.1-5 MatrixCategory_as_CategoryOfRows</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7C701DBF7BAE649A">1.2 <span class="Heading">Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7A25998E79499AE8">1.2-1 UnderlyingFieldForHomalg</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8513433F7DA84E66">1.2-2 UnderlyingMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X83D0E5BD8763C8FD">1.2-3 UnderlyingFieldForHomalg</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X81C6A5F382A1D4F5">1.2-4 Dimension</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7D03633A7D98026B">1.3 <span class="Heading">GAP Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7DAEE1D67CDFA35A">1.3-1 IsVectorSpaceMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7EF7321D794F9050">1.3-2 IsVectorSpaceObject</a></span>
</div></div>
</div>

<h3>1 <span class="Heading">Category of Matrices</span></h3>

<p><a id="X86EC0F0A78ECBC10" name="X86EC0F0A78ECBC10"></a></p>

<h4>1.1 <span class="Heading">Constructors</span></h4>

<p><a id="X79D9727280354457" name="X79D9727280354457"></a></p>

<h5>1.1-1 MatrixCategory</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatrixCategory</code>( <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a category</p>

<p>The argument is a homalg field <span class="Math">F</span>. The output is the matrix category over <span class="Math">F</span>. Objects in this category are non-negative integers. Morphisms from a non-negative integer <span class="Math">m</span> to a non-negative integer <span class="Math">n</span> are given by <span class="Math">m \times n</span> matrices.</p>

<p><a id="X7C6DB0457A656406" name="X7C6DB0457A656406"></a></p>

<h5>1.1-2 VectorSpaceMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VectorSpaceMorphism</code>( <var class="Arg">S</var>, <var class="Arg">M</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="Math">\mathrm{Hom}(S,R)</span></p>

<p>The arguments are an object <span class="Math">S</span> in the category of matrices over a homalg field <span class="Math">F</span>, a homalg matrix <span class="Math">M</span> over <span class="Math">F</span>, and another object <span class="Math">R</span> in the category of matrices over <span class="Math">F</span>. The output is the morphism <span class="Math">S \rightarrow R</span> in the category of matrices over <span class="Math">F</span> whose underlying matrix is given by <span class="Math">M</span>.</p>

<p><a id="X7EB162887B8F8708" name="X7EB162887B8F8708"></a></p>

<h5>1.1-3 VectorSpaceObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VectorSpaceObject</code>( <var class="Arg">d</var>, <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an object</p>

<p>The arguments are a non-negative integer <span class="Math">d</span> and a homalg field <span class="Math">F</span>. The output is an object in the category of matrices over <span class="Math">F</span> of dimension <span class="Math">d</span>. This function delegates to <code class="code">MatrixCategoryObject</code>.</p>

<p><a id="X818AE01E7C9D21A5" name="X818AE01E7C9D21A5"></a></p>

<h5>1.1-4 MatrixCategoryObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatrixCategoryObject</code>( <var class="Arg">cat</var>, <var class="Arg">d</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an object</p>

<p>The arguments are a matrix category <span class="Math">cat</span> over a field and a non-negative integer <span class="Math">d</span>. The output is an object in <span class="Math">cat</span> of dimension <span class="Math">d</span>.</p>

<p><a id="X7A0CAF7F7CE754E7" name="X7A0CAF7F7CE754E7"></a></p>

<h5>1.1-5 MatrixCategory_as_CategoryOfRows</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatrixCategory_as_CategoryOfRows</code>( <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a category</p>

<p>The argument is a homalg field <span class="Math">F</span>. The output is the matrix category over <span class="Math">F</span>, constructed internally as a wrapper category of the <code class="code">CategoryOfRows</code> of <span class="Math">F</span>. Only available if the package <code class="code">AdditiveClosuresForCAP</code> is available.</p>

<p><a id="X7C701DBF7BAE649A" name="X7C701DBF7BAE649A"></a></p>

<h4>1.2 <span class="Heading">Attributes</span></h4>

<p><a id="X7A25998E79499AE8" name="X7A25998E79499AE8"></a></p>

<h5>1.2-1 UnderlyingFieldForHomalg</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingFieldForHomalg</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a homalg field</p>

<p>The argument is a morphism <span class="Math">\alpha</span> in the matrix category over a homalg field <span class="Math">F</span>. The output is the field <span class="Math">F</span>.</p>

<p><a id="X8513433F7DA84E66" name="X8513433F7DA84E66"></a></p>

<h5>1.2-2 UnderlyingMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingMatrix</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a homalg matrix</p>

<p>The argument is a morphism <span class="Math">\alpha</span> in a matrix category. The output is its underlying matrix <span class="Math">M</span>.</p>

<p><a id="X83D0E5BD8763C8FD" name="X83D0E5BD8763C8FD"></a></p>

<h5>1.2-3 UnderlyingFieldForHomalg</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingFieldForHomalg</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a homalg field</p>

<p>The argument is an object <span class="Math">A</span> in the matrix category over a homalg field <span class="Math">F</span>. The output is the field <span class="Math">F</span>.</p>

<p><a id="X81C6A5F382A1D4F5" name="X81C6A5F382A1D4F5"></a></p>

<h5>1.2-4 Dimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Dimension</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a non-negative integer</p>

<p>The argument is an object <span class="Math">A</span> in a matrix category. The output is the dimension of <span class="Math">A</span>.</p>

<p><a id="X7D03633A7D98026B" name="X7D03633A7D98026B"></a></p>

<h4>1.3 <span class="Heading">GAP Categories</span></h4>

<p><a id="X7DAEE1D67CDFA35A" name="X7DAEE1D67CDFA35A"></a></p>

<h5>1.3-1 IsVectorSpaceMorphism</h5>

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

<p>The GAP category of morphisms in the category of matrices of a field <span class="Math">F</span>.</p>

<p><a id="X7EF7321D794F9050" name="X7EF7321D794F9050"></a></p>

<h5>1.3-2 IsVectorSpaceObject</h5>

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

<p>The GAP category of objects in the category of matrices of a field <span class="Math">F</span>.</p>


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