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Notebook[{

Cell[CellGroupData[{
Cell["Unitarize", "Title"],

Cell[BoxData[
    \(<< Unitarize`\)], "Input"],

Cell[CellGroupData[{

Cell["Spherical Polygon Inequalities", "Section"],

Cell["Check the sperical polygon inequalities on various values", "Text"],

Cell[CellGroupData[{

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Cell[BoxData[
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Cell["An example with tolerance", "Text"],

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Cell["Check strict equality", "Text"],

Cell[CellGroupData[{

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Cell["Unitarization", "Section"],

Cell["Functions to construct arbitrary matrices in SL2(C) and SU2", "Text"],

Cell[BoxData[{
    \(\(randomInt[] := 
        Random[Integer, {\(-10\), 10}];\)\), "\[IndentingNewLine]", 
    \(\(randomCpx[] := 
        randomInt[] + I\ randomInt[];\)\), "\[IndentingNewLine]", 
    \(randomSL2[] := 
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          x}, \[IndentingNewLine]P = {{randomCpx[], 
              randomCpx[]}, {randomCpx[], x}}; \[IndentingNewLine]P = 
          P /. \(Solve[Det[P] \[Equal] 1, 
                x]\)[\([1]\)]]\), "\[IndentingNewLine]", 
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        b = randomCpx[]; \[IndentingNewLine]P = {{a, b}, {\(-Conjugate[b]\), 
              Conjugate[a]}}; \[IndentingNewLine]P/
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Cell["\<\
Construct two simultanously unitarizable matrices M1, M2 by \
simultaneously conjugating two matrices L1, L2 in SU2 by a matrix P in \
SL2(C).\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
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    \(L2 = randomSU2[]\)}], "Input"],

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Cell[CellGroupData[{

Cell[BoxData[{
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Cell["The unitarizer", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell["Check whether M1 and M2 are simultaneously unitarizable", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Unitarizable[M1, M2]\)], "Input"],

Cell[BoxData[
    \(True\)], "Output"]
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Cell["Compute the upper triangular unitarizer Q", "Text"],

Cell[CellGroupData[{

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Cell[BoxData[
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Cell["Verify that Q is a unitarizer", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
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Cell[CellGroupData[{

Cell[BoxData[{
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    \(SU2[N2]\)}], "Input"],

Cell[BoxData[
    \(True\)], "Output"],

Cell[BoxData[
    \(True\)], "Output"]
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Cell["A numerical example with tolerence", "Text"],

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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