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

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  <div align="center"><img align="middle" src="doctitle.png"/></div>
  <hr/>
  <div align="center">Operations on Matrices</div>
  <hr/>
Axiom provides both left and right scalar multiplication.
<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>
 <li>
  <input type="submit" id="p2" class="subbut" 
    onclick="handleFree(['p1','p2']);"
    value="4*m*(-5)" />
  <div id="ansp2"><div></div></div>
 </li>
</ul>
You can add, subtract, and multiply matrices provided, of course, that the
matrices have compatible dimensions. If not, an error message is displayed.
<ul>
 <li>
  <input type="submit" id="p3" class="subbut" 
    onclick="makeRequest('p3');"
    value="n:=matrix([[1,0,-2],[-3,5,1]])" />
  <div id="ansp3"><div></div></div>
 </li>
</ul>
This following product is defined but n*m is not.
<ul>
 <li>
  <input type="submit" id="p4" class="subbut" 
    onclick="handleFree(['p1','p3','p4']);"
    value="m*n" />
  <div id="ansp4"><div></div></div>
 </li>
</ul>
The operations <a href="dbopnrows.xhtml">nrows</a> and
<a href="dbopncols.xhtml">ncols</a> return the number of rows and
columns of a matrix. You can extract a row or a column of a matrix using
the operations <a href="dboprow.xhtml">row</a> and
<a href="dbopcolumn.xhtml">column</a>. The object returned ia a
<a href="db.xhtml?Vector">Vector</a>. Here is the third column of the matrix n.
<ul>
 <li>
  <input type="submit" id="p5" class="subbut" 
    onclick="handleFree(['p3','p5']);"
    value="vec:=column(n,3)" />
  <div id="ansp5"><div></div></div>
 </li>
</ul>
You can multiply a matrix on the left by a "row vector" and on the right by
a "column vector".
<ul>
 <li>
  <input type="submit" id="p6" class="subbut" 
    onclick="handleFree(['p1','p5','p6']);"
    value="vec*m" />
  <div id="ansp6"><div></div></div>
 </li>
</ul>
The operation <a href="dbopinverse.xhtml">inverse</a> computes the inverse
of a matrix if the matrix is invertible, and returns "failed" if not. This
Hilbert matrix invertible.
<ul>
 <li>
  <input type="submit" id="p7" class="subbut" 
    onclick="makeRequest('p7');"
    value="hilb:=matrix([[1/(i+j) for i in 1..3] for j in 1..3])" />
  <div id="ansp7"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p8" class="subbut" 
    onclick="handleFree(['p7','p8']);"
    value="inverse(hilb)" />
  <div id="ansp8"><div></div></div>
 </li>
</ul>
This matrix is not invertible.
<ul>
 <li>
  <input type="submit" id="p9" class="subbut" 
    onclick="makeRequest('p9');"
    value="mm:=matrix([[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]])" />
  <div id="ansp9"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p10" class="subbut" 
    onclick="handleFree(['p9','p10']);"
    value="inverse(mm)" />
  <div id="ansp10"><div></div></div>
 </li>
</ul>
The operation <a href="dbopdeterminant.xhtml">determinant</a> computes the
determinant of a matrix provided that the entries of the matrix belong to a
<a href="db.xhtml?CommutativeRing">CommutativeRing</a>. The above matrix mm
is not invertible and, hence, must have determinant 0.
<ul>
 <li>
  <input type="submit" id="p11" class="subbut" 
    onclick="handleFree(['p9','p11']);"
    value="determinant(mm)" />
  <div id="ansp11"><div></div></div>
 </li>
</ul>
The operation <a href="dboptrace.xhtml">trace</a> computes the trace of a
square matrix.
<ul>
 <li>
  <input type="submit" id="p12" class="subbut" 
    onclick="handleFree(['p9','p12']);"
    value="trace(mm)" />
  <div id="ansp12"><div></div></div>
 </li>
</ul>
The operation <a href="dboprank.xhtml">rank</a> computes the rank of a matrix:
the maximal number of linearly independent rows or columns.
<ul>
 <li>
  <input type="submit" id="p13" class="subbut" 
    onclick="handleFree(['p9','p13']);"
    value="rank(mm)" />
  <div id="ansp13"><div></div></div>
 </li>
</ul>
The operation <a href="dbopnullity.xhtml">nullity</a> computes the nullity
of a matrix: the dimension of its null space.
<ul>
 <li>
  <input type="submit" id="p14" class="subbut" 
    onclick="handleFree(['p9','p14']);"
    value="nullity(mm)" />
  <div id="ansp14"><div></div></div>
 </li>
</ul>
The operation <a href="dbopnullspace.xhtml">nullSpace</a> returns a list 
containing a basis for the null space of a matrix. Note that the nullity is
the number of elements in a basis for the null space.
<ul>
 <li>
  <input type="submit" id="p15" class="subbut" 
    onclick="handleFree(['p9','p15']);"
    value="nullSpace(mm)" />
  <div id="ansp15"><div></div></div>
 </li>
</ul>
The operation <a href="dboprowechelon.xhtml">rowEchelon</a> returns the row
echelon form of a matrix. It is easy to see that the rank of this matrix is
two and that its nullity is also two.
<ul>
 <li>
  <input type="submit" id="p16" class="subbut" 
    onclick="handleFree(['p9','p16']);"
    value="rowEchelon(mm)" />
  <div id="ansp16"><div></div></div>
 </li>
</ul>
For more information see
<a href="axbook/section-1.6.xhtml">Expanding to Higher Dimensions</a>,
<a href="axbook/section-8.4.xhtml">
Computation of Eigenvalues and Eigenvectors</a>, and 
<a href="axbook/section-9.27.xhtml#subsec-9.27.4">
An Example: Determinant of a Hilbert Matrix</a>. Also see
<a href="db.xhtml?Permanent">Permanent</a>,
<a href="db.xhtml?Vector">Vector</a>,
<a href="db.xhtml?OneDimensionalArray">OneDimensionalArray</a>, and
<a href="db.xhtml?TwoDimensionalArray">TwoDimensionalArray</a>. Issue the
system command
<ul>
 <li>
  <input type="submit" id="p17" class="subbut" 
    onclick="showcall('p17');"
   value=")show Matrix"/>
  <div id="ansp17"><div></div></div>
 </li>
</ul>
to display the full ist of operations defined by 
<a href="db.xhtml?Matrix">Matrix</a>.
 </body>
</html>
axiom-hypertex-data 20120501-8 / usr / share / doc / axiom-doc / hypertex / linoperations.xhtml
