<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>Matrix ** RingElement -- a binary operator, usually used for tensor product or Cartesian product</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___Matrix_sp_pl_pl_sp__Matrix.html">next</a> | <a href="___Matrix_sp_st_st_sp__Ring.html">previous</a> | <a href="___Matrix_sp_pl_pl_sp__Matrix.html">forward</a> | <a href="___Matrix_sp_st_st_sp__Ring.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>Matrix ** RingElement -- a binary operator, usually used for tensor product or Cartesian product</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>f ** r</tt></div>
</dd></dl>
</div>
</li>
<li><span>Operator: <a href="__st_st.html" title="a binary operator, usually used for tensor product or Cartesian product">**</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span></span></li>
<li><span><tt>r</tt>, <span>a <a href="___Ring__Element.html">ring element</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span>the tensor product of <tt>f</tt> with <tt>r</tt></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : f = matrix "2,3,4;5,6,7"
o1 = | 2 3 4 |
| 5 6 7 |
2 3
o1 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i2 : f ** 10
o2 = | 20 30 40 |
| 50 60 70 |
2 3
o2 : Matrix ZZ <--- ZZ</pre>
</td></tr>
</table>
<p>When the ring element is homogeneous, the degrees of the source module can change, which is what makes this operation different from scalar multiplication.</p>
<table class="examples"><tr><td><pre>i3 : QQ[x,y]
o3 = QQ[x, y]
o3 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i4 : f = matrix "x,y"
o4 = | x y |
1 2
o4 : Matrix (QQ[x, y]) <--- (QQ[x, y])</pre>
</td></tr>
<tr><td><pre>i5 : g = f ** y^7
o5 = | xy7 y8 |
1 2
o5 : Matrix (QQ[x, y]) <--- (QQ[x, y])</pre>
</td></tr>
<tr><td><pre>i6 : h = f * y^7
o6 = | xy7 y8 |
1 2
o6 : Matrix (QQ[x, y]) <--- (QQ[x, y])</pre>
</td></tr>
<tr><td><pre>i7 : degrees g
o7 = {{{0}}, {{8}, {8}}}
o7 : List</pre>
</td></tr>
<tr><td><pre>i8 : degrees h
o8 = {{{0}}, {{1}, {1}}}
o8 : List</pre>
</td></tr>
</table>
</div>
</div>
</div>
</body>
</html>
