(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 118643, 3139]*) (*NotebookOutlinePosition[ 119512, 3167]*) (* CellTagsIndexPosition[ 119468, 3163]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[{\(Off[General::spelll]\), "\[IndentingNewLine]", RowBox[{\(Print["\"]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\<1.\>"]\), "\ \[IndentingNewLine]", RowBox[{ RowBox[{"A", "=", RowBox[{"(", GridBox[{ {"3", "2", "1"}, {"1", "0", "2"}, {"4", "1", "3"} }], ")"}]}], ";"}], "\[IndentingNewLine]", \(Asym\ = \ 1\/2*\((A + Transpose[ A])\);\), "\[IndentingNewLine]", \(Ass\ = \ \ 1\/2*\((A - Transpose[ A])\);\), "\[IndentingNewLine]", \(Print[\*"\"\< A = \ \!\(1\/2\)(A+\!\(A\^T\))+\!\(1\/2\)(A-\!\(A\^T\)) = \>\"", Asym // MatrixForm, \ "\< + \>", Ass\ // MatrixForm, "\< = \>", \ Asym + Ass // MatrixForm]\), "\[IndentingNewLine]", RowBox[{\(Clear[A, Asym, Ass]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{\(Print["\<2.\>"]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"O1", "=", RowBox[{\(1\/\@2\), "*", RowBox[{"(", GridBox[{ {"1", "1"}, {\(-1\), "1"} }], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"O2", "=", RowBox[{"(", GridBox[{ {"0", \(-1\)}, {"1", "0"} }], ")"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"O3", "=", RowBox[{\(1\/2\), "*", RowBox[{"(", GridBox[{ {\(\@3\), "1"}, {"1", \(-\@3\)} }], ")"}]}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\< a)\>"]\), "\ \[IndentingNewLine]", \(O11 = O1 . 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Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\".\"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\@\(3\/2\)\/2 + 1\/\(2\ \@2\)\), \(\(-\(\@\(3\/2\)\/2\)\) + 1\/\(2\ \@2\)\)}, {\(\(-\(\@\(3\/2\)\/2\)\) + 1\/\(2\ \@2\)\), \(\(-\(\@\(3\/2\)\/2\)\) - 1\/\(2\ \@2\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\" = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], SequenceForm[ " O13 * \!\(O13\^T\) = ", MatrixForm[ {{ Plus[ Times[ Rational[ 1, 2], Power[ Rational[ 3, 2], Rational[ 1, 2]]], Times[ Rational[ 1, 2], Power[ 2, Rational[ -1, 2]]]], Plus[ Times[ Rational[ -1, 2], Power[ Rational[ 3, 2], Rational[ 1, 2]]], Times[ Rational[ 1, 2], Power[ 2, Rational[ -1, 2]]]]}, { Plus[ Times[ Rational[ -1, 2], Power[ Rational[ 3, 2], Rational[ 1, 2]]], Times[ Rational[ 1, 2], Power[ 2, Rational[ -1, 2]]]], Plus[ Times[ Rational[ -1, 2], Power[ Rational[ 3, 2], Rational[ 1, 2]]], Times[ Rational[ -1, 2], Power[ 2, Rational[ -1, 2]]]]}}], ".", MatrixForm[ {{ Plus[ Times[ Rational[ 1, 2], Power[ Rational[ 3, 2], Rational[ 1, 2]]], Times[ Rational[ 1, 2], Power[ 2, Rational[ -1, 2]]]], Plus[ Times[ Rational[ -1, 2], Power[ Rational[ 3, 2], Rational[ 1, 2]]], Times[ Rational[ 1, 2], Power[ 2, Rational[ -1, 2]]]]}, { Plus[ Times[ Rational[ -1, 2], Power[ Rational[ 3, 2], Rational[ 1, 2]]], Times[ Rational[ 1, 2], Power[ 2, Rational[ -1, 2]]]], Plus[ Times[ Rational[ -1, 2], Power[ Rational[ 3, 2], Rational[ 1, 2]]], Times[ Rational[ -1, 2], Power[ 2, Rational[ -1, 2]]]]}}], " = ", MatrixForm[ {{1, 0}, {0, 1}}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" O22 * \\!\\(O22\\^T\\) = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-1\), "0"}, {"0", \(-1\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\".\"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-1\), "0"}, {"0", \(-1\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\" = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], SequenceForm[ " O22 * \!\(O22\^T\) = ", MatrixForm[ {{-1, 0}, {0, -1}}], ".", MatrixForm[ {{-1, 0}, {0, -1}}], " = ", MatrixForm[ {{1, 0}, {0, 1}}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" O21 * \\!\\(O21\\^T\\) = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/\@2\), \(-\(1\/\@2\)\)}, {\(1\/\@2\), \(1\/\@2\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\".\"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/\@2\), \(1\/\@2\)}, {\(-\(1\/\@2\)\), \(1\/\@2\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\" = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], SequenceForm[ " O21 * \!\(O21\^T\) = ", MatrixForm[ {{ Power[ 2, Rational[ -1, 2]], Times[ -1, Power[ 2, Rational[ -1, 2]]]}, { Power[ 2, Rational[ -1, 2]], Power[ 2, Rational[ -1, 2]]}}], ".", MatrixForm[ {{ Power[ 2, Rational[ -1, 2]], Power[ 2, Rational[ -1, 2]]}, { Times[ -1, Power[ 2, Rational[ -1, 2]]], Power[ 2, Rational[ -1, 2]]}}], " = ", MatrixForm[ {{1, 0}, {0, 1}}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" O23 * \\!\\(O23\\^T\\) = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-\(1\/2\)\), \(\@3\/2\)}, {\(\@3\/2\), \(1\/2\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\".\"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-\(1\/2\)\), \(\@3\/2\)}, {\(\@3\/2\), \(1\/2\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\" = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], SequenceForm[ " O23 * \!\(O23\^T\) = ", MatrixForm[ {{ Rational[ -1, 2], Times[ Rational[ 1, 2], Power[ 3, Rational[ 1, 2]]]}, { Times[ Rational[ 1, 2], Power[ 3, Rational[ 1, 2]]], Rational[ 1, 2]}}], ".", MatrixForm[ {{ Rational[ -1, 2], Times[ Rational[ 1, 2], Power[ 3, Rational[ 1, 2]]]}, { Times[ Rational[ 1, 2], Power[ 3, Rational[ 1, 2]]], Rational[ 1, 2]}}], " = ", MatrixForm[ {{1, 0}, {0, 1}}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" O33 * \\!\\(O33\\^T\\) = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\".\"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\" = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], SequenceForm[ " O33 * \!\(O33\^T\) = ", MatrixForm[ {{1, 0}, {0, 1}}], ".", MatrixForm[ {{1, 0}, {0, 1}}], " = ", MatrixForm[ {{1, 0}, {0, 1}}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" O31 * \\!\\(O31\\^T\\) = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\@\(3\/2\)\/2 - 1\/\(2\ \@2\)\), \(\@\(3\/2\)\/2 + 1\/\(2\ \@2\)\)}, {\(\@\(3\/2\)\/2 + 1\/\(2\ \@2\)\), \(\(-\(\@\(3\/2\)\/2\)\) + 1\/\(2\ \@2\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\".\"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\@\(3\/2\)\/2 - 1\/\(2\ \@2\)\), \(\@\(3\/2\)\/2 + 1\/\(2\ \@2\)\)}, {\(\@\(3\/2\)\/2 + 1\/\(2\ \@2\)\), \(\(-\(\@\(3\/2\)\/2\)\) + 1\/\(2\ \@2\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\" = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], SequenceForm[ " O31 * \!\(O31\^T\) = ", MatrixForm[ {{ Plus[ Times[ Rational[ 1, 2], Power[ Rational[ 3, 2], Rational[ 1, 2]]], Times[ Rational[ -1, 2], Power[ 2, Rational[ -1, 2]]]], Plus[ Times[ Rational[ 1, 2], Power[ Rational[ 3, 2], Rational[ 1, 2]]], Times[ Rational[ 1, 2], Power[ 2, Rational[ -1, 2]]]]}, { Plus[ Times[ Rational[ 1, 2], Power[ Rational[ 3, 2], Rational[ 1, 2]]], Times[ 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"\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\".\"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2\), \(-\(\@3\/2\)\)}, {\(-\(\@3\/2\)\), \(-\(1\/2\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\" = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], SequenceForm[ " O32 * \!\(O32\^T\) = ", MatrixForm[ {{ Rational[ 1, 2], Times[ Rational[ -1, 2], Power[ 3, Rational[ 1, 2]]]}, { Times[ Rational[ -1, 2], Power[ 3, Rational[ 1, 2]]], Rational[ -1, 2]}}], ".", MatrixForm[ {{ Rational[ 1, 2], Times[ Rational[ -1, 2], Power[ 3, Rational[ 1, 2]]]}, { Times[ Rational[ -1, 2], Power[ 3, Rational[ 1, 2]]], Rational[ -1, 2]}}], " = ", MatrixForm[ {{1, 0}, {0, 1}}]], Editable->False]], "Print"], Cell[BoxData[ \(" b)"\)], "Print"], Cell[BoxData[ 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Rational[ -1, 2]]], Times[ Complex[ 0, 1], Power[ 2, Rational[ -1, 2]]]}}], ".", MatrixForm[ {{ Times[ -1, Power[ 2, Rational[ -1, 2]]], Times[ Complex[ 0, -1], Power[ 2, Rational[ -1, 2]]]}, { Power[ 2, Rational[ -1, 2]], Times[ Complex[ 0, -1], Power[ 2, Rational[ -1, 2]]]}}], " = ", MatrixForm[ {{1, 0}, {0, 1}}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" U23 * \\!\\(\\(U23\\^+\\)\\) = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/\@2\), \(-\(1\/\@2\)\)}, {\(-\(1\/\@2\)\), \(-\(1\/\@2\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\".\"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/\@2\), \(-\(1\/\@2\)\)}, {\(-\(1\/\@2\)\), \(-\(1\/\@2\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\" = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", 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"\[InvisibleSpace]", "\<\" = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], SequenceForm[ " U33 * \!\(\(U33\^+\)\) = ", MatrixForm[ {{-1, 0}, {0, -1}}], ".", MatrixForm[ {{-1, 0}, {0, -1}}], " = ", MatrixForm[ {{1, 0}, {0, 1}}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" U31 * \\!\\(\\(U31\\^+\\)\\) = \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2 + \[ImaginaryI]\/2\), \(\(-\(1\/2\)\) + \[ImaginaryI]\ \/2\)}, {\(\(-\(1\/2\)\) + \[ImaginaryI]\/2\), \(1\/2 + \[ImaginaryI]\ \/2\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]], "\[InvisibleSpace]", "\<\".\"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2 - \[ImaginaryI]\/2\), \(\(-\(1\/2\)\) - \[ImaginaryI]\ \/2\)}, {\(\(-\(1\/2\)\) - \[ImaginaryI]\/2\), \(1\/2 - \[ImaginaryI]\ \/2\)} }], 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