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<!DOCTYPE html> <html lang="en"> <head> <title>grenet -- Construct 2^n-1 by 2^n-1 matrix with determinant equal to the permanent of the input matrix</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Computes the permanent of a square matrix." href="index.html">Permanents</a> :: <a title="Construct 2^n-1 by 2^n-1 matrix with determinant equal to the permanent of the input matrix" href="_grenet.html">grenet</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_pminors.html">next</a> | <a href="_glynn.html">previous</a> | <a href="_pminors.html">forward</a> | <a href="_glynn.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>grenet -- Construct 2^n-1 by 2^n-1 matrix with determinant equal to the permanent of the input matrix</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">F = grenet M</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">M</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, a square matrix in any commutative ring</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">N</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, a $2^n-1\times 2^n-1$ matrix <span class="tt">N</span> with determinant equal to the permanent of <span class="tt">M</span></span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>Uses Grenet's combinatorial construction (see B. Grenet: An Upper Bound for the Permanent versus Determinant Problem (2012)).</p> <p>Here is the 7x7 matrix constructed from the 3x3 generic matrix of variables.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[vars(0..8)] o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : M = genericMatrix(R,a,3,3) o2 = | a d g | | b e h | | c f i | 3 3 o2 : Matrix R <-- R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : N = grenet M o3 = | 1 0 e 0 h 0 0 | | 0 1 b 0 0 h 0 | | 0 0 1 0 0 0 i | | 0 0 0 1 b e 0 | | 0 0 0 0 1 0 f | | 0 0 0 0 0 1 c | | a d 0 g 0 0 0 | 7 7 o3 : Matrix R <-- R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : det N o4 = c*e*g + b*f*g + c*d*h + a*f*h + b*d*i + a*e*i o4 : R</code></pre> </td> </tr> </table> <div> <p>Here is the 15x15 matrix constructed from a 4x4 generic matrix of variable (note that the even case has -1 on the diagonal).</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : R=QQ[a..p] o5 = R o5 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : M=genericMatrix(R,4,4) o6 = | a e i m | | b f j n | | c g k o | | d h l p | 4 4 o6 : Matrix R <-- R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : N = grenet M o7 = | -1 0 f 0 j 0 0 0 n 0 0 0 0 0 0 | | 0 -1 b 0 0 j 0 0 0 n 0 0 0 0 0 | | 0 0 -1 0 0 0 k 0 0 0 o 0 0 0 0 | | 0 0 0 -1 b f 0 0 0 0 0 n 0 0 0 | | 0 0 0 0 -1 0 g 0 0 0 0 0 o 0 0 | | 0 0 0 0 0 -1 c 0 0 0 0 0 0 o 0 | | 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 p | | 0 0 0 0 0 0 0 -1 b f 0 j 0 0 0 | | 0 0 0 0 0 0 0 0 -1 0 g 0 k 0 0 | | 0 0 0 0 0 0 0 0 0 -1 c 0 0 k 0 | | 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 l | | 0 0 0 0 0 0 0 0 0 0 0 -1 c g 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 h | | 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 d | | a e 0 i 0 0 0 m 0 0 0 0 0 0 0 | 15 15 o7 : Matrix R <-- R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : det N o8 = d*g*j*m + c*h*j*m + d*f*k*m + b*h*k*m + c*f*l*m + b*g*l*m + d*g*i*n + ------------------------------------------------------------------------ c*h*i*n + d*e*k*n + a*h*k*n + c*e*l*n + a*g*l*n + d*f*i*o + b*h*i*o + ------------------------------------------------------------------------ d*e*j*o + a*h*j*o + b*e*l*o + a*f*l*o + c*f*i*p + b*g*i*p + c*e*j*p + ------------------------------------------------------------------------ a*g*j*p + b*e*k*p + a*f*k*p o8 : R</code></pre> </td> </tr> </table> <div> <p>Here is the construction for a matrix of integers.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i9 : M = matrix{{1,2,3,4},{5,6,7,8},{9,10,11,12},{13,14,15,16}} o9 = | 1 2 3 4 | | 5 6 7 8 | | 9 10 11 12 | | 13 14 15 16 | 4 4 o9 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : permanents(4,M) o10 = ideal 55456 o10 : Ideal of ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : N = grenet M o11 = | -1 0 6 0 7 0 0 0 8 0 0 0 0 0 0 | | 0 -1 5 0 0 7 0 0 0 8 0 0 0 0 0 | | 0 0 -1 0 0 0 11 0 0 0 12 0 0 0 0 | | 0 0 0 -1 5 6 0 0 0 0 0 8 0 0 0 | | 0 0 0 0 -1 0 10 0 0 0 0 0 12 0 0 | | 0 0 0 0 0 -1 9 0 0 0 0 0 0 12 0 | | 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 16 | | 0 0 0 0 0 0 0 -1 5 6 0 7 0 0 0 | | 0 0 0 0 0 0 0 0 -1 0 10 0 11 0 0 | | 0 0 0 0 0 0 0 0 0 -1 9 0 0 11 0 | | 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 15 | | 0 0 0 0 0 0 0 0 0 0 0 -1 9 10 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 14 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 13 | | 1 2 0 3 0 0 0 4 0 0 0 0 0 0 0 | 15 15 o11 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : det N o12 = 55456</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> <div> <p></p> <p></p> </div> </div> <div> <h2>See also</h2> <ul> <li><span><a title="ideal generated by square permanents of a matrix" href="../../Macaulay2Doc/html/_permanents.html">permanents</a> -- ideal generated by square permanents of a matrix</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">grenet</span>:</h2> <ul> <li><kbd>grenet(Matrix)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Construct 2^n-1 by 2^n-1 matrix with determinant equal to the permanent of the input matrix" href="_grenet.html">grenet</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Permanents.m2:533:0</span>.</p> </div> </div> </div> </body> </html>
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