/usr/share/doc/axiom-doc/hypertex/linintro.xhtml

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  <div align="center"><img align="middle" src="doctitle.png"/></div>
  <hr/>
  <div align="center">Expanding to Higher Dimensions</div>
  <hr/>
To get higher dimensional aggregates, you can create one-dimensional 
aggregates with elements that are themselves aggregates, for example,
lists of list, one-dimensional arrays of list of multisets, and so on. For
applications requiring two-dimensional homogeneous aggregates, you will
likely find two-dimensional arrays and matrices useful.

The entries in <a href="db.xhtml?TwoDimensionalArray">TwoDimensionalArray</a>
and <a href="?Matrix">Matrix</a> objects are all the same type, 
except that those for <a href="db.xhtml?Matrix">Matrix</a> must belong to a
<a href="db.xhtml?Ring">Ring</a>. You create and access elements in roughly
the same way. Since matrices have an understood algebraic structure, certain
algebraic operations are available for matrices but not for arrays. Because
of this, we limit our discussion here to <a href="db.xhtml?Matrix">Matrix</a>,
that can be regarded as an extension of
<a href="db.xhtml?TwoDimensionalArray">TwoDimensionalArray</a>. See
<a href="pagetwodimensionalarray.xhtml">TwoDimensionalArray</a>
For more
information about Axiom's linear algebra facilities see
<a href="pagematrix.xhtml">Matrix</a>,
<a href="pagepermanent.xhtml">Permanent</a>,
<a href="pagesquarematrix.xhtml">SquareMatrix</a>,
<a href="pagevector.xhtml">Vector</a>,
<a href="axbook/section-8.4.xhtml">
Computation of Eigenvalues and Eigenvectors</a>, and
<a href="axbook/section-8.5.xhtml">
Solution of Linear and Polynomial Equations</a>.

You can create a matrix from a list of lists, where each of the inner
lists represents a row of the matrix.
<ul>
 <li>
  <input type="submit" id="p1" class="subbut" 
    onclick="makeRequest('p1');"
    value="m:=matrix([[1,2],[3,4]])" />
  <div id="ansp1"><div></div></div>
 </li>
</ul>
The "collections" construct (see
<a href="axbook/section-5.5.xhtml">
Creating Lists and Streams with Iterators</a>)
is useful for creating matrices whose entries are given by formulas.
<ul>
 <li>
  <input type="submit" id="p2" class="subbut" 
    onclick="makeRequest('p2');"
    value="matrix([[1/(i+j-x) for i in 1..4] for j in 1..4])" />
  <div id="ansp2"><div></div></div>
 </li>
</ul>
Let vm denote the three by three Vandermonde matrix.
<ul>
 <li>
  <input type="submit" id="p3" class="subbut" 
    onclick="makeRequest('p3');"
    value="vm:=matrix [[1,1,1],[x,y,z],[x*x,y*y,z*z]]" />
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 </li>
</ul>
Use this syntax to extract an entry in the matrix.
<ul>
 <li>
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    onclick="handleFree(['p3','p4']);"
    value="vm(3,3)" />
  <div id="ansp4"><div></div></div>
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</ul>
You can also pull out a <a href="dboprow.xhtml">row</a> or a column.
<ul>
 <li>
  <input type="submit" id="p5" class="subbut" 
    onclick="handleFree(['p3','p5']);"
    value="column(vm,2)" />
  <div id="ansp5"><div></div></div>
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</ul>
You can do arithmetic.
<ul>
 <li>
  <input type="submit" id="p6" class="subbut" 
    onclick="handleFree(['p3','p6']);"
    value="vm*vm" />
  <div id="ansp6"><div></div></div>
 </li>
</ul>
You can perform operations such as 
<a href="dboptranspose.xhtml">transpose</a>,
<a href="dboptrace.xhtml">trace</a>, and
<a href="dbopdeterminant.xhtml">determinant</a>
<ul>
 <li>
  <input type="submit" id="p7" class="subbut" 
    onclick="handleFree(['p3','p7']);"
    value="factor determinant vm" />
  <div id="ansp7"><div></div></div>
 </li>
</ul>
 </body>
</html>
axiom-hypertex-data 20120501-8 / usr / share / doc / axiom-doc / hypertex / linintro.xhtml
