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<!DOCTYPE html> <html lang="en"> <head> <title>Kronecker -- Kronecker and rational normal forms</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="Kronecker and rational normal forms" href="index.html">Kronecker</a> :: <a title="Kronecker and rational normal forms" href="index.html">Kronecker</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="_decompose__Graded__Module.html">next</a> | previous | <a href="_decompose__Graded__Module.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Kronecker -- Kronecker and rational normal forms</h1> <div> <h2>Description</h2> This package provides methods for computing the Kronecker and rational normal forms of a matrix. </div> <div> <div> <div> <h2>Author</h2> <ul> <li>Edward Carter<span> <<a href="mailto:edward.carter%40gmail.com">edward.carter@gmail.com</a>></span></li> </ul> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>0.3.3</b> of Kronecker, released <b>July 16, 2010</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@misc{KroneckerSource, title = {{Kronecker: Kronecker and rational normal forms. Version~0.3.3}}, author = {Edward Carter}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Functions and commands <ul> <li><span><a title="decompose a graded module over a ring of graded module maps" href="_decompose__Graded__Module.html">decomposeGradedModule</a> -- decompose a graded module over a ring of graded module maps</span></li> <li><span><a title="decompose a module into a direct sum of simple modules" href="_decompose__Module.html">decomposeModule</a> -- decompose a module into a direct sum of simple modules</span></li> <li><span><span class="tt">doubleDualMap</span> (missing documentation)<!--tag: doubleDualMap--> </span></li> <li><span><a title="data which classify a matrix pencil up to strict equivalence" href="_kronecker__Indices.html">kroneckerIndices</a> -- data which classify a matrix pencil up to strict equivalence</span></li> <li><span><a title="Kronecker normal form of a matrix of linear forms" href="_kronecker__Normal__Form.html">kroneckerNormalForm</a> -- Kronecker normal form of a matrix of linear forms</span></li> <li><span><a title="rational normal form of a matrix" href="_rational__Normal__Form.html">rationalNormalForm</a> -- rational normal form of a matrix</span></li> </ul> </li> <li>Methods <ul> <li><span><kbd>decomposeGradedModule(GradedModuleMap,GradedModuleMap)</kbd> -- see <span><a title="decompose a graded module over a ring of graded module maps" href="_decompose__Graded__Module.html">decomposeGradedModule</a> -- decompose a graded module over a ring of graded module maps</span></span></li> <li><span><kbd>decomposeModule(Module)</kbd> -- see <span><a title="decompose a module into a direct sum of simple modules" href="_decompose__Module.html">decomposeModule</a> -- decompose a module into a direct sum of simple modules</span></span></li> <li><span><span class="tt">doubleDualMap(Module,Module)</span> (missing documentation)<!--tag: (doubleDualMap,Module,Module)--> </span></li> <li><span><kbd>kroneckerIndices(Matrix,RingElement,RingElement)</kbd> -- see <span><a title="data which classify a matrix pencil up to strict equivalence" href="_kronecker__Indices.html">kroneckerIndices</a> -- data which classify a matrix pencil up to strict equivalence</span></span></li> <li><span><a title="data classifying the normal form of a pair of matrices" href="_kronecker__Indices_lp__Matrix_cm__Matrix_rp.html">kroneckerIndices(Matrix,Matrix)</a> -- data classifying the normal form of a pair of matrices</span></li> <li><span><kbd>kroneckerNormalForm(Matrix)</kbd> -- see <span><a title="Kronecker normal form of a matrix of linear forms" href="_kronecker__Normal__Form.html">kroneckerNormalForm</a> -- Kronecker normal form of a matrix of linear forms</span></span></li> <li><span><kbd>kroneckerNormalForm(Matrix,RingElement,RingElement)</kbd> -- see <span><a title="Kronecker normal form of a matrix of linear forms" href="_kronecker__Normal__Form.html">kroneckerNormalForm</a> -- Kronecker normal form of a matrix of linear forms</span></span></li> <li><span><a title="normal form of a pair of matrices of scalars" href="_kronecker__Normal__Form_lp__Matrix_cm__Matrix_rp.html">kroneckerNormalForm(Matrix,Matrix)</a> -- normal form of a pair of matrices of scalars</span></li> <li><span><kbd>rationalNormalForm(Matrix)</kbd> -- see <span><a title="rational normal form of a matrix" href="_rational__Normal__Form.html">rationalNormalForm</a> -- rational normal form of a matrix</span></span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Kronecker and rational normal forms" href="index.html">Kronecker</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">Kronecker.m2</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Kronecker.m2:1053:0</span>.</p> </div> </div> </div> </body> </html>