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<head><title>minors(ZZ,Matrix) -- ideal generated by minors</title>
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<div><h1>minors(ZZ,Matrix) -- ideal generated by minors</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>minors(n,M)</tt></div>
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<li><span>Function: <a href="_minors_lp__Z__Z_cm__Matrix_rp.html" title="ideal generated by minors">minors</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>n</tt>, <span>an <a href="___Z__Z.html">integer</a></span>, order of the minor</span></li>
<li><span><tt>M</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a map between free modules</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Ideal.html">ideal</a></span>, the ideal generated by the <tt>n</tt> by <tt>n</tt> minors of the matrix <tt>M</tt></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>First => </tt><span><span>a <a href="___List.html">list</a></span>, <span>default value null</span>, if given, should be a list of two integer lists, which will be the first minor computed</span></span></li>
<li><span><tt>Limit => </tt><span><span>an <a href="___Z__Z.html">integer</a></span>, <span>default value infinity</span>, the maximum number of minors to find</span></span></li>
<li><span><a href="_minors_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, -- choose between Bareiss and Cofactor algorithms</span></li>
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<div class="single"><h2>Description</h2>
<div>Minors are generated in the same order as that used by <a href="_subsets_lp__Z__Z_cm__Z__Z_rp.html" title="produce all the subsets">subsets(ZZ,ZZ)</a>.<table class="examples"><tr><td><pre>i1 : R = ZZ[a..f];</pre>
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<tr><td><pre>i2 : M = matrix{{a,b,c},{d,e,f}}
o2 = | a b c |
| d e f |
2 3
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : minors(2,M)
o3 = ideal (- b*d + a*e, - c*d + a*f, - c*e + b*f)
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : minors(2,M,Limit=>1)
o4 = ideal(- b*d + a*e)
o4 : Ideal of R</pre>
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<p>When <tt>n</tt> is negative, the unit ideal is returned, to preserve the expected ordering among the resulting ideals.</p>
<table class="examples"><tr><td><pre>i5 : minors(1,M)
o5 = ideal (a, d, b, e, c, f)
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : minors(0,M)
o6 = ideal 1
o6 : Ideal of R</pre>
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<tr><td><pre>i7 : minors(-1,M)
o7 = ideal 1
o7 : Ideal of R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_exterior__Power.html" title="exterior power">exteriorPower</a> -- exterior power</span></li>
<li><span><a href="_determinant.html" title="determinant of a matrix">determinant</a> -- determinant of a matrix</span></li>
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