(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 14470, 482] NotebookOptionsPosition[ 12742, 419] NotebookOutlinePosition[ 13077, 434] CellTagsIndexPosition[ 13034, 431] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell["Here' s the matrix in question", "Text", CellChangeTimes->{{3.437133669810506*^9, 3.437133677188142*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"A", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"2", ",", "2", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "2", ",", "2"}], "}"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.437133693554194*^9, 3.437133712920387*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"2", "2", "0"}, { RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}]}, {"0", "2", "2"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.437133706949394*^9, 3.437133713630061*^9}}] }, Open ]], Cell["\<\ First I find the characteristic polynomial of A and its roots\ \>", "Text", CellChangeTimes->{{3.43713372184964*^9, 3.437133739127016*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"charpoly", " ", "=", " ", RowBox[{"Det", "[", RowBox[{"A", "-", RowBox[{"t", " ", RowBox[{"IdentityMatrix", "[", "3", "]"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.437133747277055*^9, 3.43713376605491*^9}}], Cell[BoxData[ RowBox[{"16", "-", RowBox[{"16", " ", "t"}], "+", RowBox[{"6", " ", SuperscriptBox["t", "2"]}], "-", SuperscriptBox["t", "3"]}]], "Output", CellChangeTimes->{{3.437133761733968*^9, 3.437133766668458*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"charpoly", "\[Equal]", "0"}], ",", "t"}], "]"}]], "Input", CellChangeTimes->{{3.437133775408218*^9, 3.437133781161297*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"t", "\[Rule]", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"t", "\[Rule]", RowBox[{"2", "-", RowBox[{"2", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", "\[Rule]", RowBox[{"2", "+", RowBox[{"2", " ", "\[ImaginaryI]"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.437133782140401*^9}] }, Open ]], Cell["\<\ Then I find eigenvectors for the first two of these roots (I can ignore the \ third, because it's the complex conjugate of the second)\ \>", "Text", CellChangeTimes->{{3.437133786743823*^9, 3.437133819616112*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"A", ".", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3"}], "}"}]}], "\[Equal]", RowBox[{"2", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3"}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.437133827596948*^9, 3.437133840770045*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Solve", "::", "\<\"svars\"\>"}], RowBox[{ ":", " "}], "\<\"Equations may not give solutions for all \\\"solve\\\" \ variables. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\ \\\", ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/Solve/svars\\\", \ ButtonNote -> \\\"Solve::svars\\\"]\\)\"\>"}]], "Message", "MSG", CellChangeTimes->{3.437133841901279*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"x2", "\[Rule]", "0"}], ",", RowBox[{"x1", "\[Rule]", RowBox[{"-", "x3"}]}]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.437133841908664*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{"A", ".", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3"}], "}"}]}], "\[Equal]", RowBox[{ RowBox[{"(", RowBox[{"2", "-", RowBox[{"2", "I"}]}], ")"}], RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3"}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.437133843062027*^9, 3.437133862062228*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Solve", "::", "\<\"svars\"\>"}], RowBox[{ ":", " "}], "\<\"Equations may not give solutions for all \\\"solve\\\" \ variables. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\ \\\", ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/Solve/svars\\\", \ ButtonNote -> \\\"Solve::svars\\\"]\\)\"\>"}]], "Message", "MSG", CellChangeTimes->{3.437133862502696*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"x1", "\[Rule]", "x3"}], ",", RowBox[{"x2", "\[Rule]", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", "x3"}]}]}], "}"}], "}"}]], "Output",\ CellChangeTimes->{3.437133862655272*^9}] }, Open ]], Cell["\<\ So (-1, 0, 1) is a basis for the eigenspace of the eigenvalue 2, and (1, -i, \ 1) is a basis for the eigenspace of the eigenvalue 2 - 2 i. The real and \ imaginary parts of the latter vector are (1,0,1) and (0,1,0). With these two \ vectors and the eigenvector for 2, I get a basis for R^3.\ \>", "Text", CellChangeTimes->{{3.437133871719363*^9, 3.437133994437722*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"basis", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "1"}], ",", "0"}], "}"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.437133997263219*^9, 3.437134018250827*^9}, 3.437134278018516*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "1"}], ",", "0"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.437134092484234*^9, 3.437134278998435*^9}] }, Open ]], Cell["\<\ And the change of coordinates matrix from this basis to the standard basis is \ just the matrix whose columns are the basis vectors. Since I' ve entered the \ basis vectors as rows, I take the transpose\ \>", "Text", CellChangeTimes->{{3.437134022313884*^9, 3.437134069924434*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"chg", " ", "=", " ", RowBox[{"Transpose", "[", "basis", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.437134078211607*^9, 3.437134089214414*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"-", "1"}], "1", "0"}, {"0", "0", RowBox[{"-", "1"}]}, {"1", "1", "0"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.437134089661524*^9, 3.437134095229976*^9}, 3.437134281386621*^9}] }, Open ]], Cell["\<\ According to the theorem in class, the matrix for my linear transformation \ should be nicer if I use this basis :\ \>", "Text", CellChangeTimes->{{3.437134106995989*^9, 3.437134168336507*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"newA", " ", "=", " ", RowBox[{ RowBox[{"Inverse", "[", "chg", "]"}], ".", "A", ".", "chg"}]}], "]"}]], "Input", CellChangeTimes->{{3.437134174973778*^9, 3.437134197613423*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"2", "0", "0"}, {"0", "2", RowBox[{"-", "2"}]}, {"0", "2", "2"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.437134190829893*^9, 3.437134198033072*^9}, 3.437134283746368*^9}] }, Open ]], Cell["\<\ So newA has ' block diagonal' form, as I hoped. Moreover, the bottom block \ can be rewritten as \ \>", "Text", CellChangeTimes->{{3.437134202163822*^9, 3.437134255147916*^9}, { 3.437134327590345*^9, 3.437134356647828*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"block", " ", "=", " ", RowBox[{"r", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "Theta", "]"}], ",", RowBox[{"-", RowBox[{"Sin", "[", "Theta", "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Sin", "[", "Theta", "]"}], ",", RowBox[{"Cos", "[", "Theta", "]"}]}], "}"}]}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.437134306579626*^9, 3.437134315562776*^9}, { 3.437134364094308*^9, 3.437134421724299*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"r", " ", RowBox[{"Cos", "[", "Theta", "]"}]}], RowBox[{ RowBox[{"-", "r"}], " ", RowBox[{"Sin", "[", "Theta", "]"}]}]}, { RowBox[{"r", " ", RowBox[{"Sin", "[", "Theta", "]"}]}], RowBox[{"r", " ", RowBox[{"Cos", "[", "Theta", "]"}]}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.43713441490045*^9, 3.437134422141974*^9}}] }, Open ]], Cell["\<\ where r and Theta are determined by the eigenvalue 2-2I = r e^(-i Theta). \ That is, r = | 2 - 2 i | = 2^(3/2) is the magnitude of 2-2I, and Theta = -tan^{-1}( 2/(-2)) = Pi/4 is minus the argument of 2-2I. Just to be sure, I check it:\ \>", "Text", CellChangeTimes->{{3.437134424323314*^9, 3.437134708934589*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"block", "//.", RowBox[{"{", RowBox[{ RowBox[{"r", "\[Rule]", RowBox[{"2", RowBox[{"Sqrt", "[", "2", "]"}]}]}], ",", RowBox[{"Theta", "\[Rule]", RowBox[{"Pi", "/", "4"}]}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.437134710974004*^9, 3.43713473641987*^9}, { 3.437134767884884*^9, 3.437134772197239*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"2", RowBox[{"-", "2"}]}, {"2", "2"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.437134737533943*^9, 3.437134772594559*^9}] }, Open ]], Cell["\<\ ... which agrees with the bottom - right block in the matrix above\ \>", "Text", CellChangeTimes->{{3.437134774449179*^9, 3.437134797596981*^9}}] }, WindowSize->{640, 623}, WindowMargins->{{150, Automatic}, {Automatic, 52}}, FrontEndVersion->"6.0 for Linux x86 (32-bit) (April 20, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[568, 21, 112, 1, 31, "Text"], Cell[CellGroupData[{ Cell[705, 26, 446, 13, 32, "Input"], Cell[1154, 41, 725, 21, 73, "Output"] }, Open ]], Cell[1894, 65, 150, 3, 31, "Text"], Cell[CellGroupData[{ Cell[2069, 72, 254, 6, 32, "Input"], Cell[2326, 80, 233, 6, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2596, 91, 185, 4, 32, "Input"], Cell[2784, 97, 420, 13, 31, "Output"] }, Open ]], Cell[3219, 113, 224, 4, 51, "Text"], Cell[CellGroupData[{ Cell[3468, 121, 325, 9, 32, "Input"], Cell[3796, 132, 421, 8, 24, "Message"], Cell[4220, 142, 225, 7, 31, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4482, 154, 398, 12, 32, "Input"], Cell[4883, 168, 421, 8, 24, "Message"], Cell[5307, 178, 267, 9, 31, "Output"] }, Open ]], Cell[5589, 190, 382, 6, 71, "Text"], Cell[CellGroupData[{ Cell[5996, 200, 439, 13, 32, "Input"], Cell[6438, 215, 366, 11, 31, "Output"] }, Open ]], Cell[6819, 229, 293, 5, 71, "Text"], Cell[CellGroupData[{ Cell[7137, 238, 205, 4, 32, "Input"], Cell[7345, 244, 751, 22, 73, "Output"] }, Open ]], Cell[8111, 269, 204, 4, 51, "Text"], Cell[CellGroupData[{ Cell[8340, 277, 242, 6, 32, "Input"], Cell[8585, 285, 729, 21, 73, "Output"] }, Open ]], Cell[9329, 309, 237, 5, 51, "Text"], Cell[CellGroupData[{ Cell[9591, 318, 583, 17, 55, "Input"], Cell[10177, 337, 935, 27, 59, "Output"] }, Open ]], Cell[11127, 367, 334, 8, 111, "Text"], Cell[CellGroupData[{ Cell[11486, 379, 409, 11, 32, "Input"], Cell[11898, 392, 669, 19, 57, "Output"] }, Open ]], Cell[12582, 414, 156, 3, 31, "Text"] } ] *) (* End of internal cache information *)