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Notebook[{
Cell["\t ", "Text"],

Cell[CellGroupData[{

Cell["Matrix Exterior Differential Calculus: Examples", "Subtitle",
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Cell[TextData[{
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  StyleBox["\n",
    FontWeight->"Bold"],
  StyleBox["Institute of Nuclear Physics, ",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox[ButtonBox["NCSR \"Demokritos\"",
    ButtonData:>{
      URL[ "http://www.demokritos.gr/"], None},
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  StyleBox[", \nAghia Paraskevi 15310, Greece. ",
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Cell["\<\
\tWarning: The first four examples are not independent. They must \
be evaluated in the order they appear.\
\>", "Text",
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  FontColor->RGBColor[0, 0, 1]],

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Cell[TextData[{
  StyleBox["Example I:",
    FontWeight->"Bold"],
  "\tElementary calculations with symbolic matrices"
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  "\tI.e., ",
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  " will stand for a symmetric 0-form matrix  and ",
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  ", ",
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  "  for an 1-form matrix and its transpose. "
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  "\tNote that the identity matrix multiplying the last term is not written \
explicitly. ",
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  "  is a global variable, used in the definition of ",
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  ", that can be assigned by the user. "
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  "\t",
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  " must be used to multiply matrices together , or with scalar forms,"
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Cell[BoxData[
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Cell[BoxData[
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  "\tThis is correct, since  ",
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  " has applied trace identities in obtaining this expression"
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  "\tApplying the Cayley-Hamilton theorem"
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Cell[BoxData[
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  "\tThen, the Cayley-Hamilton theorem gives (",
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  " is understood to multiply the identity matrix)"
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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  "\t(",
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  " takes into account that ",
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  " ",
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  " multiplies the 3-dim identity matrix)"
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  FontColor->RGBColor[0, 0, 1]],

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Cell[BoxData[
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Cell[BoxData[
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      s3\ d[S]\[Wedge]S + 2\ s2\ d[s2]\[Wedge]S + d[s3]\[Wedge]S\[Wedge]S\)], 
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Cell[BoxData[
    \(0\)], "Output"]
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Cell[TextData[{
  StyleBox["Example III:",
    FontWeight->"Bold"],
  " \tA composite calculation"
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Cell[TextData[{
  "\tOne can assign ",
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  ", say as a linear combination of ",
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  " and the traceless part of ",
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  " with 1-form coefficients"
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Cell[CellGroupData[{

Cell[BoxData[
    \(d[S] = q[1]\[Wedge]S + q[2]\[Wedge]\((S\[Wedge]S - 2  s2/3)\)\)], 
  "Input"],

Cell[BoxData[
    \(\(-\(2\/3\)\)\ s2\ q[2] + q[1]\[Wedge]S + q[2]\[Wedge]S\[Wedge]S\)], 
  "Output"]
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Cell[TextData[{
  "\t(Note that, even though S is a 0-form matrix, one must use ",
  StyleBox["Wedge", "Output"],
  " to multiply it with scalar differential forms). Then ",
  StyleBox["d", "Output"],
  " uses this definition when applicable"
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  FontColor->RGBColor[0, 0, 1]],

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Cell[BoxData[
    \(d[S\[Wedge]S]\)], "Input"],

Cell[BoxData[
    \(2\ s3\ q[2] + 2\/3\ s2\ q[2]\[Wedge]S + 2\ q[1]\[Wedge]S\[Wedge]S\)], 
  "Output"]
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  "\tNow, from the definition of ",
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  FontColor->RGBColor[0, 0, 1]],

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Cell[BoxData[
    \(d[s2] = tr[S\[Wedge]d[S]]\)], "Input"],

Cell[BoxData[
    \(2\ s2\ q[1] + 3\ s3\ q[2]\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(d[s3] = tr[S\[Wedge]S\[Wedge]d[S]]\)], "Input"],

Cell[BoxData[
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Cell[TextData[{
  "\tThe integrability conditions for ",
  StyleBox["d[s2],", "Output"],
  " ",
  StyleBox["d[s3]", "Output"],
  " imply that ",
  StyleBox["q[1]", "Output"],
  " and ",
  StyleBox["q[2]", "Output"],
  " must make the following two equations vanish:"
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Cell[BoxData[
    \(eq2 = d[%%]\)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(SolveAlways[{eq1 == 0, eq2 == 0}, {d[q[_]]}]\)], "Input"],

Cell[BoxData[
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        d[q[1]] \[Rule] 0, d[q[2]] \[Rule] \(-\((q[1]\[Wedge]q[2])\)\)}, {
        d[q[1]] \[Rule] 0, d[q[2]] \[Rule] \(-\((q[1]\[Wedge]q[2])\)\)}, {
        d[q[1]] \[Rule] 0, d[q[2]] \[Rule] \(-\((q[1]\[Wedge]q[2])\)\)}, {
        d[q[2]] \[Rule] \(-\((q[1]\[Wedge]q[2])\)\), d[q[1]] \[Rule] 0}, {
        d[q[2]] \[Rule] \(-\(\(\@3\ d[q[1]]\)\/\@s2\)\) - q[1]\[Wedge]q[2], 
        s3 \[Rule] \(2\ s2\^\(3/2\)\)\/\(3\ \@3\)}, {
        d[q[2]] \[Rule] \(\@3\ d[q[1]]\)\/\@s2 - q[1]\[Wedge]q[2], 
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  "\tThus, we can assign to  ",
  StyleBox["d[q[i]]", "Output"],
  " the values:"
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Cell[BoxData[
    \(d[q[1]] = 0; d[q[2]] = \(-\((q[1]\[Wedge]q[2])\)\); \)], "Input"],

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  "\tThen the integrability condition for our choice of ",
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  " is satisfied:"
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    \(0\)], "Output"]
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Cell[TextData[{
  StyleBox["Example IV: ",
    FontWeight->"Bold"],
  "\tConverting to component form - vectors"
}], "Subsubtitle",
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Cell["\<\
\tTo obtain the component form of any of these equations, one must \
define a substitution rule for  S, s2, s3 consistent with the assumptions \
made. For example, let\
\>", "Text",
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  FontColor->RGBColor[0, 0, 1]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Scomp = Array[Sc, {3, 3}]\)], "Input"],

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Cell[CellGroupData[{

Cell[BoxData[
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      s2 \[Rule] 
        Sc[1, 1]\^2 + Sc[1, 2]\^2 + Sc[1, 3]\^2 + Sc[1, 1]\ Sc[2, 2] + 
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      s3 \[Rule] 
        Sc[1, 1]\ Sc[1, 2]\^2 - Sc[1, 1]\^2\ Sc[2, 2] + 
          Sc[1, 2]\^2\ Sc[2, 2] - Sc[1, 3]\^2\ Sc[2, 2] - 
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Cell[TextData[{
  "\t",
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  " is equivalent to ",
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  " when acting on an expression that does not contain a hidden identity \
matrix:"
}], "Text",
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  FontColor->RGBColor[0, 0, 1]],

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Cell[BoxData[
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Cell[TextData[{
  "\tBut ",
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              Sc[2, 2]\^2 + Sc[2, 3]\^2)\), 
        2\/3\ \((
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              Sc[2, 2]\^2 + Sc[2, 3]\^2)\)}, {
        2\/3\ \((
            Sc[1, 1]\^2 + Sc[1, 2]\^2 + Sc[1, 3]\^2 + Sc[1, 1]\ Sc[2, 2] + 
              Sc[2, 2]\^2 + Sc[2, 3]\^2)\), 0, 
        2\/3\ \((
            Sc[1, 1]\^2 + Sc[1, 2]\^2 + Sc[1, 3]\^2 + Sc[1, 1]\ Sc[2, 2] + 
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        2\/3\ \((
            Sc[1, 1]\^2 + Sc[1, 2]\^2 + Sc[1, 3]\^2 + Sc[1, 1]\ Sc[2, 2] + 
              Sc[2, 2]\^2 + Sc[2, 3]\^2)\), 
        2\/3\ \((
            Sc[1, 1]\^2 + Sc[1, 2]\^2 + Sc[1, 3]\^2 + Sc[1, 1]\ Sc[2, 2] + 
              Sc[2, 2]\^2 + Sc[2, 3]\^2)\), 0}}\)], "Output"]
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Cell[BoxData[
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Cell[TextData[{
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  StyleBox["d[S]", "Output"],
  " is, of course, equal to ",
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  FontSize->14,
  FontColor->RGBColor[0, 0, 1]],

Cell[BoxData[
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Cell[TextData[{
  "\tNow we can evaluate the component form of the the integrability \
condition for",
  StyleBox[" d[S]", "Output"]
}], "Text",
  FontSize->14,
  FontColor->RGBColor[0, 0, 1]],

Cell[CellGroupData[{

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Cell[BoxData[
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          1\/3\ Sc[1, 2]\^2\ q[1]\[Wedge]q[2] + 
          1\/3\ Sc[1, 3]\^2\ q[1]\[Wedge]q[2] - 
          2\/3\ Sc[1, 1]\ Sc[2, 2]\ q[1]\[Wedge]q[2] - 
          2\/3\ Sc[2, 2]\^2\ q[1]\[Wedge]q[2] - 
          2\/3\ Sc[2, 3]\^2\ q[1]\[Wedge]q[2] - 
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            q[1]\[Wedge]q[2], 
        Sc[1, 1]\ Sc[1, 2]\ q[1]\[Wedge]q[2] + 
          Sc[1, 2]\ Sc[2, 2]\ q[1]\[Wedge]q[2] + 
          Sc[1, 3]\ Sc[2, 3]\ q[1]\[Wedge]q[2] - 
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            q[1]\[Wedge]q[2], 
        \(-Sc[1, 3]\)\ Sc[2, 2]\ q[1]\[Wedge]q[2] + 
          Sc[1, 2]\ Sc[2, 3]\ q[1]\[Wedge]q[2] - 
          \((\(-Sc[1, 3]\)\ Sc[2, 2] + Sc[1, 2]\ Sc[2, 3])\)\ 
            q[1]\[Wedge]q[2]}, {
        Sc[1, 1]\ Sc[1, 2]\ q[1]\[Wedge]q[2] + 
          Sc[1, 2]\ Sc[2, 2]\ q[1]\[Wedge]q[2] + 
          Sc[1, 3]\ Sc[2, 3]\ q[1]\[Wedge]q[2] - 
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        \(-\(2\/3\)\)\ Sc[1, 1]\^2\ q[1]\[Wedge]q[2] + 
          1\/3\ Sc[1, 2]\^2\ q[1]\[Wedge]q[2] - 
          2\/3\ Sc[1, 3]\^2\ q[1]\[Wedge]q[2] - 
          2\/3\ Sc[1, 1]\ Sc[2, 2]\ q[1]\[Wedge]q[2] + 
          1\/3\ Sc[2, 2]\^2\ q[1]\[Wedge]q[2] + 
          1\/3\ Sc[2, 3]\^2\ q[1]\[Wedge]q[2] - 
          1\/3\ \((
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                2\ Sc[1, 1]\ Sc[2, 2] + Sc[2, 2]\^2 + Sc[2, 3]\^2)\)\ 
            q[1]\[Wedge]q[2], 
        Sc[1, 2]\ Sc[1, 3]\ q[1]\[Wedge]q[2] - 
          Sc[1, 1]\ Sc[2, 3]\ q[1]\[Wedge]q[2] - 
          \((Sc[1, 2]\ Sc[1, 3] - Sc[1, 1]\ Sc[2, 3])\)\ q[1]\[Wedge]q[2]}, {
        \(-Sc[1, 3]\)\ Sc[2, 2]\ q[1]\[Wedge]q[2] + 
          Sc[1, 2]\ Sc[2, 3]\ q[1]\[Wedge]q[2] - 
          \((\(-Sc[1, 3]\)\ Sc[2, 2] + Sc[1, 2]\ Sc[2, 3])\)\ 
            q[1]\[Wedge]q[2], 
        Sc[1, 2]\ Sc[1, 3]\ q[1]\[Wedge]q[2] - 
          Sc[1, 1]\ Sc[2, 3]\ q[1]\[Wedge]q[2] - 
          \((Sc[1, 2]\ Sc[1, 3] - Sc[1, 1]\ Sc[2, 3])\)\ q[1]\[Wedge]q[2], 
        1\/3\ Sc[1, 1]\^2\ q[1]\[Wedge]q[2] - 
          2\/3\ Sc[1, 2]\^2\ q[1]\[Wedge]q[2] + 
          1\/3\ Sc[1, 3]\^2\ q[1]\[Wedge]q[2] + 
          4\/3\ Sc[1, 1]\ Sc[2, 2]\ q[1]\[Wedge]q[2] + 
          1\/3\ Sc[2, 2]\^2\ q[1]\[Wedge]q[2] + 
          1\/3\ Sc[2, 3]\^2\ q[1]\[Wedge]q[2] + 
          1\/3\ \((
              Sc[1, 1]\^2 + Sc[1, 2]\^2 + Sc[1, 3]\^2 - 
                2\ Sc[1, 1]\ Sc[2, 2] - 2\ Sc[2, 2]\^2 - 2\ Sc[2, 3]\^2)\)\ 
            q[1]\[Wedge]q[2] + 
          1\/3\ \((
              \(-2\)\ Sc[1, 1]\^2 + Sc[1, 2]\^2 - 2\ Sc[1, 3]\^2 - 
                2\ Sc[1, 1]\ Sc[2, 2] + Sc[2, 2]\^2 + Sc[2, 3]\^2)\)\ 
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[TextData[{
  StyleBox["Example V:",
    FontWeight->"Bold"],
  " \tGauge Transformation of Connection and Curvature"
}], "Subsubtitle",
  FontSize->16],

Cell[TextData[{
  "\tIn this example, the gauge transformation formulae for the connection \
1-forms and the curvature 2-forms are given in matrix form. The notation \
follows the conventions of:\n\t R. Penrose and W. Rindler,  ",
  StyleBox["Spinors and Space-Time",
    FontSlant->"Italic"],
  ", (Cambridge U. Press, 1984) vol 1, section 4.13.\n\t The formulation in \
terms of 2- and 3-dim matrices is developed in:\n\t  S. Bonanos,  ",
  StyleBox[
  "The vacuum Einstein equations as closed ideals of matrix-valued \
differential forms",
    FontSlant->"Italic"],
  ", Journal of Mathematical Physics ",
  StyleBox["32",
    FontWeight->"Bold"],
  ", 188 (1991) ",
  "\n",
  "\t"
}], "Text",
  FontSize->14],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    \("OK"\)], "Print"]
}, Open  ]],

Cell[TextData[
"\tG is a gauge transformation matrix, Ginv its inverse and \[Omega] is the \
connection 1-form matrix."], "Text",
  FontSize->14,
  FontColor->RGBColor[0, 0, 1]],

Cell[CellGroupData[{

Cell[BoxData[
    \(DeclareForms[{1, \[Omega]0, \[Omega]1, \[Omega]2}]\)], "Input"],

Cell[BoxData[
    \("OK"\)], "Print"]
}, Open  ]],

Cell[TextData[{
  "\tThe 1-forms ",
  StyleBox["\[Omega]0,", "Output"],
  " ",
  StyleBox["\[Omega]1,", "Output"],
  " ",
  StyleBox["\[Omega]2", "Output"],
  " are the components of the connection matrix ",
  StyleBox["\[Omega]", "Output"],
  ". In the two-dim representation, the matrices ",
  StyleBox["G,", "Output"],
  " ",
  StyleBox["\[Omega]", "Output"],
  " have the form:"
}], "Text",
  FontSize->14,
  FontColor->RGBColor[0, 0, 1]],

Cell[BoxData[
    \(G[2] = {{a, b}, {c, d}}/Sqrt[a\ d - b\ c]; \n
    \[Omega][2]
         = {{\ \[Omega]1, \(-\[Omega]0\)}, {\[Omega]2, \(-\ \[Omega]1\)}}; 
    \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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                 {\(a\/\@\(\(-b\)\ c + a\ d\)\), 
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                 {\(c\/\@\(\(-b\)\ c + a\ d\)\), 
                  \(d\/\@\(\(-b\)\ c + a\ d\)\)}
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        "\[InvisibleSpace]", 
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          Times[ a, 
            Power[ 
              Plus[ 
                Times[ -1, b, c], 
                Times[ a, d]], 
              Rational[ -1, 2]]], 
          Times[ b, 
            Power[ 
              Plus[ 
                Times[ -1, b, c], 
                Times[ a, d]], 
              Rational[ -1, 2]]]}, {
          Times[ c, 
            Power[ 
              Plus[ 
                Times[ -1, b, c], 
                Times[ a, d]], 
              Rational[ -1, 2]]], 
          Times[ d, 
            Power[ 
              Plus[ 
                Times[ -1, b, c], 
                Times[ a, d]], 
              Rational[ -1, 2]]]}}], " ", 1, " ", 
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          Times[ -1, \[Omega]0]}, {\[Omega]2, 
          Times[ -1, \[Omega]1]}}]],
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}, Open  ]],

Cell[TextData[{
  "\tIn the three-dim representation, the matrices ",
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  " ",
  StyleBox["\[Omega]", "Output"],
  " have the form:"
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  FontSize->14,
  FontColor->RGBColor[0, 0, 1]],

Cell[BoxData[
    \(G[3]
         = {{\((a\^2 - b\^2 - c\^2 + d\^2)\)/2, 
            \(-I\) \((a\^2 + b\^2 - c\^2 - d\^2)\)/2, \(-a\)\ b + c\ d}, {
            I \((a\^2 - b\^2 + c\^2 - d\^2)\)/2, 
            \((a\^2 + b\^2 + c\^2 + d\^2)\)/2, \(-I\) \((a\ b + c\ d)\)}, {
            \(-a\)\ c + b\ d, I \((a\ c + b\ d)\), a\ d + b\ c}}/
        \((a\ d - b\ c)\); \n
    \[Omega][3]
         = {{0, \(-2\)\ I\ \[Omega]1, \[Omega]0 + \[Omega]2}, {
          2\ I\ \[Omega]1, 0, I\ \((\[Omega]0 - \[Omega]2)\)}, {
          \(-\[Omega]0\) - \[Omega]2, \(-I\)\ \((\[Omega]0 - \[Omega]2)\), 
          0}}; \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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                 {\(\(a\^2 - b\^2 - c\^2 + d\^2\)\/\(2\ 
                        \((\(-b\)\ c + a\ d)\)\)\), 
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                          \((\(-b\)\ c + a\ d)\)\)\)\), 
                  \(\(\(-a\)\ b + c\ d\)\/\(\(-b\)\ c + a\ d\)\)},
               
                 {\(\(I\ \((a\^2 - b\^2 + c\^2 - d\^2)\)\)\/\(2\ 
                        \((\(-b\)\ c + a\ d)\)\)\), 
                  \(\(a\^2 + b\^2 + c\^2 + d\^2\)\/\(2\ 
                        \((\(-b\)\ c + a\ d)\)\)\), 
                  \(-\(\(I\ \((a\ b + c\ d)\)\)\/\(\(-b\)\ c + a\ d\)\)\)},
               
                 {\(\(\(-a\)\ c + b\ d\)\/\(\(-b\)\ c + a\ d\)\), 
                  \(\(I\ \((a\ c + b\ d)\)\)\/\(\(-b\)\ c + a\ d\)\), 
                  \(\(b\ c + a\ d\)\/\(\(-b\)\ c + a\ d\)\)}
                }], ")"}],
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                  \(I\ \((\[Omega]0 - \[Omega]2)\)\)},
               
                 {\(\(-\[Omega]0\) - \[Omega]2\), 
                  \(\(-I\)\ \((\[Omega]0 - \[Omega]2)\)\), "0"}
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          Times[ 
            Rational[ 1, 2], 
            Power[ 
              Plus[ 
                Times[ -1, b, c], 
                Times[ a, d]], -1], 
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              Power[ a, 2], 
              Times[ -1, 
                Power[ b, 2]], 
              Times[ -1, 
                Power[ c, 2]], 
              Power[ d, 2]]], 
          Times[ 
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              Rational[ -1, 2]], 
            Power[ 
              Plus[ 
                Times[ -1, b, c], 
                Times[ a, d]], -1], 
            Plus[ 
              Power[ a, 2], 
              Power[ b, 2], 
              Times[ -1, 
                Power[ c, 2]], 
              Times[ -1, 
                Power[ d, 2]]]], 
          Times[ 
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            Plus[ 
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              Times[ c, d]]]}, {
          Times[ 
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            Power[ 
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                Power[ d, 2]]]], 
          Times[ 
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            Power[ 
              Plus[ 
                Times[ -1, b, c], 
                Times[ a, d]], -1], 
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              Power[ a, 2], 
              Power[ b, 2], 
              Power[ c, 2], 
              Power[ d, 2]]], 
          Times[ 
            Complex[ 0, -1], 
            Power[ 
              Plus[ 
                Times[ -1, b, c], 
                Times[ a, d]], -1], 
            Plus[ 
              Times[ a, b], 
              Times[ c, d]]]}, {
          Times[ 
            Power[ 
              Plus[ 
                Times[ -1, b, c], 
                Times[ a, d]], -1], 
            Plus[ 
              Times[ -1, a, c], 
              Times[ b, d]]], 
          Times[ 
            Complex[ 0, 1], 
            Power[ 
              Plus[ 
                Times[ -1, b, c], 
                Times[ a, d]], -1], 
            Plus[ 
              Times[ a, c], 
              Times[ b, d]]], 
          Times[ 
            Power[ 
              Plus[ 
                Times[ -1, b, c], 
                Times[ a, d]], -1], 
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              Times[ b, c], 
              Times[ a, d]]]}}], " ", 1, " ", 
        MatrixForm[ {{0, 
          Times[ 
            Complex[ 0, -2], \[Omega]1], 
          Plus[ \[Omega]0, \[Omega]2]}, {
          Times[ 
            Complex[ 0, 2], \[Omega]1], 0, 
          Times[ 
            Complex[ 0, 1], 
            Plus[ \[Omega]0, 
              Times[ -1, \[Omega]2]]]}, {
          Plus[ 
            Times[ -1, \[Omega]0], 
            Times[ -1, \[Omega]2]], 
          Times[ 
            Complex[ 0, -1], 
            Plus[ \[Omega]0, 
              Times[ -1, \[Omega]2]]], 0}}]],
      Editable->False]], "Print"]
}, Open  ]],

Cell[TextData[{
  "\tThe replacement rules ",
  StyleBox["matComp[n]", "Output"],
  " (n=2 or 3) transform the symbolic matrices ",
  StyleBox["G,", "Output"],
  " ",
  StyleBox["Ginv,", "Output"],
  " ",
  StyleBox["\[Omega]", "Output"],
  " to their respective component form"
}], "Text",
  FontSize->14,
  FontColor->RGBColor[0, 0, 1]],

Cell[BoxData[
    \(\(matComp[n_] := 
      \(matComp[n] = {G -> G[n], Gt -> Transpose[G[n]], 
          \[Omega] -> \[Omega][n], \[Omega]t -> Transpose[\[Omega][n]], 
          Ginv -> Together[Inverse[G[n]]], 
          Ginvt -> Transpose[Together[Inverse[G[n]]]]}\); \)\)], "Input"],

Cell[TextData[{
  "\tDefinition of the curvature 2-form matrix ",
  StyleBox["\[CapitalOmega]", "Output"],
  " in terms of the connection 1-form matrix ",
  StyleBox["\[Omega]", "Output"]
}], "Text",
  FontSize->14,
  FontColor->RGBColor[0, 0, 1]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[TextData[{
  "\tComponent form of ",
  StyleBox["\[CapitalOmega]", "Output"],
  ".  For brevity of notation, define ",
  StyleBox["compForm", "Output"]
}], "Text",
  FontSize->14,
  FontColor->RGBColor[0, 0, 1]],

Cell[BoxData[
    \(\(compForm[x_, n_] := toComponents[x, matComp[n]]; \)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(MatrixForm[compForm[\[CapitalOmega], 2]]\)], "Input"],

Cell[BoxData[
    TagBox[
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             {\(d[\[Omega]1] + \[Omega]0\[Wedge]\[Omega]2\), 
              \(\(-d[\[Omega]0]\) - 2\ \[Omega]0\[Wedge]\[Omega]1\)},
           
             {\(d[\[Omega]2] + 2\ \[Omega]1\[Wedge]\[Omega]2\), 
              \(\(-d[\[Omega]1]\) - \[Omega]0\[Wedge]\[Omega]2\)}
            }], ")"}],
      (MatrixForm[ #]&)]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(MatrixForm[Factor[compForm[\[CapitalOmega], 3]]]\)], "Input"],

Cell[BoxData[
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n=3, ~130 sec! (All timings refer to an Apple Macintosh G3/900 computer \
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