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<head><title>adjoint(Matrix,Module,Module) -- an adjoint map</title>
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<div><h1>adjoint(Matrix,Module,Module) -- an adjoint map</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>adjoint(f,F,G)</tt></div>
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<li><span>Function: <a href="_adjoint_lp__Matrix_cm__Module_cm__Module_rp.html" title="an adjoint map">adjoint</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a homomorphism <tt>F ** G --> H</tt> between free modules</span></li>
<li><span><tt>F</tt>, <span>a <a href="___Module.html">module</a></span>, a free module</span></li>
<li><span><tt>G</tt>, <span>a <a href="___Module.html">module</a></span>, a free module</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, the adjoint homomorphism <tt>F --> (dual G) ** H</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>All modules should be free modules over the same base ring, and the rank of the source of <tt>f</tt> should be the product of the ranks of <tt>F</tt> and <tt>G</tt>. Recall that <tt>**</tt> refers to the tensor product of modules, and that <tt>dual G</tt> is a free module with the same rank as <tt>G</tt>.<p/>
No computation is required. The resulting matrix has the same entries as <tt>f</tt>, but in a different layout.<table class="examples"><tr><td><pre>i1 : R = QQ[x_1 .. x_24];</pre>
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<tr><td><pre>i2 : f = genericMatrix(R,2,4*3)
o2 = | x_1 x_3 x_5 x_7 x_9 x_11 x_13 x_15 x_17 x_19 x_21 x_23 |
| x_2 x_4 x_6 x_8 x_10 x_12 x_14 x_16 x_18 x_20 x_22 x_24 |
2 12
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : g = adjoint(f,R^4,R^3)
o3 = | x_1 x_7 x_13 x_19 |
| x_2 x_8 x_14 x_20 |
| x_3 x_9 x_15 x_21 |
| x_4 x_10 x_16 x_22 |
| x_5 x_11 x_17 x_23 |
| x_6 x_12 x_18 x_24 |
6 4
o3 : Matrix R <--- R</pre>
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If <tt>f</tt> is homogeneous, and <tt>source f == F ** G</tt>, including the grading, then the resulting matrix will be homogeneous.<table class="examples"><tr><td><pre>i4 : g = adjoint(f,R^4,R^{-1,-1,-1})
o4 = {-1} | x_1 x_7 x_13 x_19 |
{-1} | x_2 x_8 x_14 x_20 |
{-1} | x_3 x_9 x_15 x_21 |
{-1} | x_4 x_10 x_16 x_22 |
{-1} | x_5 x_11 x_17 x_23 |
{-1} | x_6 x_12 x_18 x_24 |
6 4
o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : isHomogeneous g
o5 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_adjoint1_lp__Matrix_cm__Module_cm__Module_rp.html" title="an adjoint map">adjoint1</a> -- an adjoint map</span></li>
<li><span><a href="_flip_lp__Module_cm__Module_rp.html" title="matrix of commutativity of tensor product">flip</a> -- matrix of commutativity of tensor product</span></li>
<li><span><a href="_reshape_lp__Module_cm__Module_cm__Matrix_rp.html" title="reshape a matrix">reshape</a> -- reshape a matrix</span></li>
<li><span><a href="___Module_sp_st_st_sp__Module.html" title="tensor product">Module ** Module</a> -- tensor product</span></li>
<li><span><a href="_dual.html" title="dual module or map">dual</a> -- dual module or map</span></li>
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