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<!DOCTYPE html> <html lang="en"> <head> <title>NCMatrix -- Type of a matrix over a noncommutative ring</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 href="index.html">NCAlgebra</a> :: <a title="Type of a matrix over a noncommutative ring" href="___N__C__Matrix.html">NCMatrix</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="_nc__Matrix.html">next</a> | <a href="_nc__Map.html">previous</a> | <a href="_nc__Matrix.html">forward</a> | <a href="_nc__Map.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>NCMatrix -- Type of a matrix over a noncommutative ring</h1> <div> <h2>Description</h2> <div> <p>This is the type of a matrix over a noncommutative ring. These represent homomorphisms between two free modules in chosen bases (whether you think of it as a map of left or right modules is up you). Modules themselves are not implemented yet in the <a href="index.html">NCAlgebra</a> package, but are slated for a later release.</p> </div> <div> <p>Common ways to make (and use) a matrix include</p> </div> <ul> <li><span><a title="Create an NCMatrix" href="_nc__Matrix.html">ncMatrix(List)</a> -- Create an NCMatrix</span></li> <li><span><a title="Returns a basis of an NCRing in a particular degree." href="_basis_lp__Z__Z_cm__N__C__Ring_rp.html">basis(ZZ,NCRing)</a> -- Returns a basis of an NCRing in a particular degree.</span></li> <li><span><a title="Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman" href="_right__Kernel.html">rightKernel(NCMatrix,ZZ)</a> -- Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman</span></li> <li><span><a title="Methods for computing kernels of matrices over noncommutative rings using Bergman" href="_right__Kernel__Bergman.html">rightKernelBergman(NCMatrix)</a> -- Methods for computing kernels of matrices over noncommutative rings using Bergman</span></li> </ul> <div> <p>Common ways to get information about matrices</p> </div> <ul> <li><span><a title="Gives the ring of the NCMatrix" href="_ring_lp__N__C__Matrix_rp.html">ring(NCMatrix)</a> -- Gives the ring of the NCMatrix</span></li> <li><span><a title="Returns the entries of the NCMatrix" href="_entries_lp__N__C__Matrix_rp.html">entries(NCMatrix)</a> -- Returns the entries of the NCMatrix</span></li> </ul> <div> <p>Common operations on matrices:</p> </div> <ul> <li><span><a title="Add NCMatrices" href="___N__C__Matrix_sp_pl_sp__N__C__Matrix.html">NCMatrix + NCMatrix</a> -- Add NCMatrices</span></li> <li><span><a title="Subtract NCMatrices" href="___N__C__Matrix_sp-_sp__N__C__Matrix.html">NCMatrix - NCMatrix</a> -- Subtract NCMatrices</span></li> <li><span><a title="Reduces the entries of an NCMatrix with respect to an NCGroebnerBasis" href="___N__C__Matrix_sp_pc_sp__N__C__Groebner__Basis.html">NCMatrix % NCGroebnerBasis</a> -- Reduces the entries of an NCMatrix with respect to an NCGroebnerBasis</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__N__C__Matrix.html">NCMatrix * NCMatrix</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__N__C__Ring__Element.html">NCMatrix * NCRingElement</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__Ring__Element.html">NCMatrix * RingElement</a> -- Product of NCMatrices</span></li> <li><span><a title="Factor one map through another" href="___N__C__Matrix_sp_sl_sl_sp__N__C__Matrix.html">NCMatrix // NCMatrix</a> -- Factor one map through another</span></li> <li><span><a title="Select some columns of an NCMatrix" href="___N__C__Matrix_sp_us_sp__List.html">NCMatrix _ List</a> -- Select some columns of an NCMatrix</span></li> <li><span><a title="Test equality of matrices" href="___N__C__Matrix_sp_eq_eq_sp__N__C__Matrix.html">NCMatrix == NCMatrix</a> -- Test equality of matrices</span></li> <li><span><a title="Join NCMatrices horizontally" href="___N__C__Matrix_sp_vb_sp__N__C__Matrix.html">NCMatrix | NCMatrix</a> -- Join NCMatrices horizontally</span></li> <li><span><a title="Join NCMatrices vertically" href="___N__C__Matrix_sp_vb_vb_sp__N__C__Matrix.html">NCMatrix || NCMatrix</a> -- Join NCMatrices vertically</span></li> <li><span><a title="Select some rows of an NCMatrix" href="___N__C__Matrix_sp%5E_sp__List.html">NCMatrix ^ List</a> -- Select some rows of an NCMatrix</span></li> <li><span><a title="Exponentiate an NCMatrix" href="___N__C__Matrix_sp%5E_sp__Z__Z.html">NCMatrix ^ ZZ</a> -- Exponentiate an NCMatrix</span></li> </ul> <div> <p>This is the type of a matrix with entries in an NCRing. Many of the basic operations one can perform on a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">Matrix</a> are also allowed with an <a title="Type of a matrix over a noncommutative ring" href="___N__C__Matrix.html">NCMatrix</a>, and the behavior of the functions should be similar to the corresponding 'usual' command. Some examples of creating and using NCMatrices are given below.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : A = QQ{a,b,c,d} o1 = A o1 : NCPolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : M = ncMatrix {{a,b,c,d}} o2 = | a b c d | o2 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : N = ncMatrix {{M,2*M,3*M},{4*M,5*M,6*M}} o3 = | a b c d 2*a 2*b 2*c 2*d 3*a 3*b 3*c 3*d | | 4*a 4*b 4*c 4*d 5*a 5*b 5*c 5*d 6*a 6*b 6*c 6*d | o3 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : B = QQ{x,y,z} o4 = B o4 : NCPolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : f = y*z + z*y - x^2 2 o5 = zy+yz-x o5 : B</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : g = x*z + z*x - y^2 2 o6 = zx-y +xz o6 : B</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : h = z^2 - x*y - y*x 2 o7 = z -yx-xy o7 : B</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : I = ncIdeal {f,g,h} 2 2 2 o8 = Two-sided ideal {zy+yz-x , zx-y +xz, z -yx-xy} o8 : NCIdeal</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : Igb = ncGroebnerBasis I --Calling Bergman for NCGB calculation. Complete! 2 2 2 o9 = y x-xy ; Lead Term = (y x, 1) 2 2 2 yx -x y; Lead Term = (yx , 1) 2 zx-y +xz; Lead Term = (zx, 1) 2 zy+yz-x ; Lead Term = (zy, 1) 2 2 z -yx-xy; Lead Term = (z , 1) o9 : NCGroebnerBasis</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : M = ncMatrix {{x, y, z}} o10 = | x y z | o10 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : sigma = ncMap(B,B,{y,z,x}) o11 = NCRingMap B <--- B o11 : NCRingMap</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : N = ncMatrix {{M},{sigma M}, {sigma sigma M}} o12 = | x y z | | y z x | | z x y | o12 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : Nred = N^3 % Igb o13 = | -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3 2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3 | | y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3 2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3 -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y | | 2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3 -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3 | o13 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i14 : C = B/I o14 = C o14 : NCQuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i15 : phi = ncMap(C,B,gens C) o15 = NCRingMap C <--- B o15 : NCRingMap</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i16 : NC = phi N o16 = | x y z | | y z x | | z x y | o16 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i17 : N3C = NC^3 o17 = | -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3 2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3 | | y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3 2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3 -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y | | 2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3 -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3 | o17 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i18 : X = NC + 3*NC o18 = | 4*x 4*y 4*z | | 4*y 4*z 4*x | | 4*z 4*x 4*y | o18 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i19 : Y = NC | 2*NC o19 = | x y z 2*x 2*y 2*z | | y z x 2*y 2*z 2*x | | z x y 2*z 2*x 2*y | o19 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i20 : Z = X || NC o20 = | 4*x 4*y 4*z | | 4*y 4*z 4*x | | 4*z 4*x 4*y | | x y z | | y z x | | z x y | o20 : NCMatrix</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Functions and methods returning an object of class NCMatrix:</h2> <ul> <li><span><a title="Returns a basis of an NCIdeal in a particular degree." href="_basis_lp__Z__Z_cm__N__C__Ideal_rp.html">basis(ZZ,NCIdeal)</a> -- Returns a basis of an NCIdeal in a particular degree.</span></li> <li><span><a title="Returns a basis of an NCLeftIdeal in a particular degree." href="_basis_lp__Z__Z_cm__N__C__Left__Ideal_rp.html">basis(ZZ,NCLeftIdeal)</a> -- Returns a basis of an NCLeftIdeal in a particular degree.</span></li> <li><span><a title="Returns a basis of an NCRightIdeal in a particular degree." href="_basis_lp__Z__Z_cm__N__C__Right__Ideal_rp.html">basis(ZZ,NCRightIdeal)</a> -- Returns a basis of an NCRightIdeal in a particular degree.</span></li> <li><span><a title="Returns a basis of an NCRing in a particular degree." href="_basis_lp__Z__Z_cm__N__C__Ring_rp.html">basis(ZZ,NCRing)</a> -- Returns a basis of an NCRing in a particular degree.</span></li> <li><span><a title="Lifts an NCMatrix" href="_lift_lp__N__C__Matrix_rp.html">lift(NCMatrix)</a> -- Lifts an NCMatrix</span></li> </ul> <h2>Methods that use an object of class NCMatrix:</h2> <ul> <li><span><a title="Negates NCMatrices" href="_-_sp__N__C__Matrix.html">- NCMatrix</a> -- Negates NCMatrices</span></li> <li><span><kbd>assignDegrees(NCMatrix)</kbd> -- see <span><a title="Weights entries of a matrix to make associated map of free modules graded" href="_assign__Degrees.html">assignDegrees</a> -- Weights entries of a matrix to make associated map of free modules graded</span></span></li> <li><span><kbd>assignDegrees(NCMatrix,List,List)</kbd> -- see <span><a title="Weights entries of a matrix to make associated map of free modules graded" href="_assign__Degrees.html">assignDegrees</a> -- Weights entries of a matrix to make associated map of free modules graded</span></span></li> <li><span><a title="Returns the entries of the NCMatrix" href="_entries_lp__N__C__Matrix_rp.html">entries(NCMatrix)</a> -- Returns the entries of the NCMatrix</span></li> <li><span><a title="Compute a graded component of Hom(M,N)" href="___Hom_lp__Z__Z_cm__N__C__Matrix_cm__N__C__Matrix_rp.html">Hom(ZZ,NCMatrix,NCMatrix)</a> -- Compute a graded component of Hom(M,N)</span></li> <li><span><kbd>isHomogeneous(NCMatrix)</kbd> -- see <span><a title="Determines whether the input defines a homogeneous object" href="_is__Homogeneous_lp__N__C__Ideal_rp.html">isHomogeneous(NCIdeal)</a> -- Determines whether the input defines a homogeneous object</span></span></li> <li><span><a title="Product of NCMatrices" href="___Matrix_sp_st_sp__N__C__Matrix.html">Matrix * NCMatrix</a> -- Product of NCMatrices</span></li> <li><span><a title="Reduces the entries of an NCMatrix with respect to an NCGroebnerBasis" href="___N__C__Matrix_sp_pc_sp__N__C__Groebner__Basis.html">NCMatrix % NCGroebnerBasis</a> -- Reduces the entries of an NCMatrix with respect to an NCGroebnerBasis</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__Matrix.html">NCMatrix * Matrix</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__N__C__Matrix.html">NCMatrix * NCMatrix</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__N__C__Ring__Element.html">NCMatrix * NCRingElement</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__Q__Q.html">NCMatrix * QQ</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__Ring__Element.html">NCMatrix * RingElement</a> -- Product of NCMatrices</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Matrix_sp_st_sp__Z__Z.html">NCMatrix * ZZ</a> -- Product of NCMatrices</span></li> <li><span><span class="tt">NCMatrix ** NCMatrix</span> (missing documentation)<!--tag: (**,NCMatrix,NCMatrix)--> </span></li> <li><span><a title="Add NCMatrices" href="___N__C__Matrix_sp_pl_sp__N__C__Matrix.html">NCMatrix + NCMatrix</a> -- Add NCMatrices</span></li> <li><span><span class="tt">NCMatrix ++ NCMatrix</span> (missing documentation)<!--tag: (++,NCMatrix,NCMatrix)--> </span></li> <li><span><a title="Subtract NCMatrices" href="___N__C__Matrix_sp-_sp__N__C__Matrix.html">NCMatrix - NCMatrix</a> -- Subtract NCMatrices</span></li> <li><span><a title="Factor one map through another" href="___N__C__Matrix_sp_sl_sl_sp__N__C__Matrix.html">NCMatrix // NCMatrix</a> -- Factor one map through another</span></li> <li><span><a title="Test equality of matrices" href="___N__C__Matrix_sp_eq_eq_sp__N__C__Matrix.html">NCMatrix == NCMatrix</a> -- Test equality of matrices</span></li> <li><span><kbd>NCMatrix == ZZ</kbd> -- see <span><a title="Test equality of matrices" href="___N__C__Matrix_sp_eq_eq_sp__N__C__Matrix.html">NCMatrix == NCMatrix</a> -- Test equality of matrices</span></span></li> <li><span><kbd>ZZ == NCMatrix</kbd> -- see <span><a title="Test equality of matrices" href="___N__C__Matrix_sp_eq_eq_sp__N__C__Matrix.html">NCMatrix == NCMatrix</a> -- Test equality of matrices</span></span></li> <li><span><a title="Select some rows of an NCMatrix" href="___N__C__Matrix_sp%5E_sp__List.html">NCMatrix ^ List</a> -- Select some rows of an NCMatrix</span></li> <li><span><a title="Exponentiate an NCMatrix" href="___N__C__Matrix_sp%5E_sp__Z__Z.html">NCMatrix ^ ZZ</a> -- Exponentiate an NCMatrix</span></li> <li><span><a title="Select some columns of an NCMatrix" href="___N__C__Matrix_sp_us_sp__List.html">NCMatrix _ List</a> -- Select some columns of an NCMatrix</span></li> <li><span><a title="Induced map in homogeneous degree n" href="___N__C__Matrix_sp_us_sp__Z__Z.html">NCMatrix _ ZZ</a> -- Induced map in homogeneous degree n</span></li> <li><span><a title="Join NCMatrices horizontally" href="___N__C__Matrix_sp_vb_sp__N__C__Matrix.html">NCMatrix | NCMatrix</a> -- Join NCMatrices horizontally</span></li> <li><span><a title="Join NCMatrices vertically" href="___N__C__Matrix_sp_vb_vb_sp__N__C__Matrix.html">NCMatrix || NCMatrix</a> -- Join NCMatrices vertically</span></li> <li><span><a title="Graded shift of an NCMatrix." href="___N__C__Matrix_sp__Array.html">NCMatrix Array</a> -- Graded shift of an NCMatrix.</span></li> <li><span><a title="Product of NCMatrices" href="___N__C__Ring__Element_sp_st_sp__N__C__Matrix.html">NCRingElement * NCMatrix</a> -- Product of NCMatrices</span></li> <li><span><kbd>NCRingMap NCMatrix</kbd> -- see <span><a title="Apply an NCRingMap to an element or matrix" href="___N__C__Ring__Map_sp__N__C__Ring__Element.html">NCRingMap NCRingElement</a> -- Apply an NCRingMap to an element or matrix</span></span></li> <li><span><a title="Product of NCMatrices" href="___Q__Q_sp_st_sp__N__C__Matrix.html">QQ * NCMatrix</a> -- Product of NCMatrices</span></li> <li><span><a title="Compute the resolution of coker M as a map of free right modules" href="_resolution_lp__N__C__Matrix_rp.html">resolution(NCMatrix)</a> -- Compute the resolution of coker M as a map of free right modules</span></li> <li><span><kbd>rightKernel(NCMatrix,ZZ)</kbd> -- see <span><a title="Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman" href="_right__Kernel.html">rightKernel</a> -- Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman</span></span></li> <li><span><kbd>rightKernelBergman(NCMatrix)</kbd> -- see <span><a title="Methods for computing kernels of matrices over noncommutative rings using Bergman" href="_right__Kernel__Bergman.html">rightKernelBergman</a> -- Methods for computing kernels of matrices over noncommutative rings using Bergman</span></span></li> <li><span><a title="Gives the ring of the NCMatrix" href="_ring_lp__N__C__Matrix_rp.html">ring(NCMatrix)</a> -- Gives the ring of the NCMatrix</span></li> <li><span><a title="Product of NCMatrices" href="___Ring__Element_sp_st_sp__N__C__Matrix.html">RingElement * NCMatrix</a> -- Product of NCMatrices</span></li> <li><span><a title="Transposes an NCMatrix" href="_transpose_lp__N__C__Matrix_rp.html">transpose(NCMatrix)</a> -- Transposes an NCMatrix</span></li> <li><span><a title="Product of NCMatrices" href="___Z__Z_sp_st_sp__N__C__Matrix.html">ZZ * NCMatrix</a> -- Product of NCMatrices</span></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Type of a matrix over a noncommutative ring" href="___N__C__Matrix.html">NCMatrix</a> is <span>a <a title="the class of all mutable types" href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a title="the class of all mutable hash tables" href="../../Macaulay2Doc/html/___Mutable__Hash__Table.html">MutableHashTable</a> < <a title="the class of all hash tables" href="../../Macaulay2Doc/html/___Hash__Table.html">HashTable</a> < <a title="the class of all things" href="../../Macaulay2Doc/html/___Thing.html">Thing</a>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">NCAlgebra/NCAlgebraDoc.m2:430:0</span>.</p> </div> </div> </div> </body> </html>