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<head><title>contract(Matrix,Matrix) -- contract a matrix by a matrix</title>
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<div><h1>contract(Matrix,Matrix) -- contract a matrix by a matrix</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>h = contract(m,n)</tt></div>
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<li><span>Function: <a href="_contract.html" title="contract one matrix by another">contract</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>m</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a map <tt>m : F <--- P</tt> between free modules of ranks f and p.</span></li>
<li><span><tt>n</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a map <tt>n : G <--- Q</tt> between free modules of ranks g and q.</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>h</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, the contraction of <tt>n</tt> by <tt>m</tt>, a matrix with the shape <tt>h : dual F ** G <--- dual P ** Q</tt>, whose entry in the slot <tt>h<sub>g*i+j,q*k+l</sub></tt> is the result of contracting <tt>n<sub>j,l</sub></tt>, by <tt>m<sub>i,k</sub></tt></span></li>
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<div class="single"><h2>Description</h2>
<div>The arguments can also be ring elements or vectors.<table class="examples"><tr><td><pre>i1 : R = ZZ[x,y,z]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : f = vars R ** vars R
o2 = | x2 xy xz xy y2 yz xz yz z2 |
1 9
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : contract(transpose vars R, f)
o3 = {1} | x y z y 0 0 z 0 0 |
{1} | 0 x 0 x y z 0 z 0 |
{1} | 0 0 x 0 0 y x y z |
3 9
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : contract(x, f)
o4 = | x y z y 0 0 z 0 0 |
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o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : contract(y, f)
o5 = | 0 x 0 x y z 0 z 0 |
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o5 : Matrix R <--- R</pre>
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<tr><td><pre>i6 : contract(z, f)
o6 = | 0 0 x 0 0 y x y z |
1 9
o6 : Matrix R <--- R</pre>
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This function is identical to <a href="_diff_lp__Matrix_cm__Matrix_rp.html" title="differentiate a matrix by a matrix">diff(Matrix,Matrix)</a>, except that the multiplication by integers that occurs during differentiation is omitted.<p/>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_contract_sq_lp__Matrix_cm__Matrix_rp.html" title="contract a matrix by a matrix, the dual notion">contract'</a> -- contract a matrix by a matrix, the dual notion</span></li>
<li><span><a href="_diff_spand_spcontract.html" title="">diff and contract</a></span></li>
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