(* 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[ 49728, 1559] NotebookOptionsPosition[ 45134, 1401] NotebookOutlinePosition[ 45951, 1431] CellTagsIndexPosition[ 45781, 1424] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Some sample Jordan normal form and matrix exponential calculations\ \>", "Title", CellChangeTimes->{{3.402126793823285*^9, 3.40212691650108*^9}, { 3.402127067991959*^9, 3.402127191093029*^9}, {3.4021272489713287`*^9, 3.402127271777747*^9}, {3.402224709998932*^9, 3.402224721830627*^9}}, FontSize->12], Cell[CellGroupData[{ Cell["Definitions", "Subsection", CellChangeTimes->{{3.402126793823285*^9, 3.40212691650108*^9}, { 3.402127067991959*^9, 3.402127191093029*^9}, {3.4021272489713287`*^9, 3.402127271777747*^9}, {3.40222470063172*^9, 3.402224703116126*^9}}, FontSize->12], Cell[BoxData[ RowBox[{ RowBox[{"s", "[", RowBox[{"a_", ",", "\[Lambda]_"}], "]"}], ":=", RowBox[{"a", "-", RowBox[{"\[Lambda]", " ", RowBox[{"IdentityMatrix", "[", RowBox[{"Length", "[", "a", "]"}], "]"}]}]}]}]], "Input", CellChangeTimes->{{3.402083063726235*^9, 3.402083082542369*^9}}], Cell[TextData[{ "LinearSolve, Solve and MatrixExp are built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " functions. Here are short specs of their syntax:" }], "Text", CellChangeTimes->{{3.402126793823285*^9, 3.40212691650108*^9}, { 3.402127067991959*^9, 3.402127191093029*^9}, {3.4021272489713287`*^9, 3.402127271777747*^9}, {3.402224781438066*^9, 3.402224835412966*^9}}, FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "LinearSolve"}]], "Input", CellChangeTimes->{{3.402224840190175*^9, 3.4022248452555847`*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"LinearSolve\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"m\\\", \\\"TI\\\"], \\\",\\\", StyleBox[\\\"b\\\", \ \\\"TI\\\"]}], \\\"]\\\"}]\\) finds an \\!\\(\\*StyleBox[\\\"x\\\", \ \\\"TI\\\"]\\) which solves the matrix equation \ \\!\\(\\*RowBox[{RowBox[{StyleBox[\\\"m\\\", \\\"TI\\\"], \\\".\\\", \ StyleBox[\\\"x\\\", \\\"TI\\\"]}], \\\"==\\\", StyleBox[\\\"b\\\", \ \\\"TI\\\"]}]\\). \\n\\!\\(\\*RowBox[{\\\"LinearSolve\\\", \\\"[\\\", \ StyleBox[\\\"m\\\", \\\"TI\\\"], \\\"]\\\"}]\\) generates a \\!\\(\\*RowBox[{\ \\\"LinearSolveFunction\\\", \\\"[\\\", StyleBox[\\\"\[Ellipsis]\\\", \ \\\"TR\\\"], \\\"]\\\"}]\\) which can be applied repeatedly to different \ \\!\\(\\*StyleBox[\\\"b\\\", \\\"TI\\\"]\\). \"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/LinearSolve"]}]], "Print", "PrintUsage", CellChangeTimes->{3.402224849090035*^9}, CellTags->"Info3402199648-9239799"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Solve"}]], "Input", CellChangeTimes->{{3.402224840190175*^9, 3.4022248551584587`*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"Solve\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"eqns\\\", \\\"TI\\\"], \\\",\\\", StyleBox[\\\"vars\\\", \ \\\"TI\\\"]}], \\\"]\\\"}]\\) attempts to solve an equation or set of \ equations for the variables \\!\\(\\*StyleBox[\\\"vars\\\", \\\"TI\\\"]\\). \ \\n\\!\\(\\*RowBox[{\\\"Solve\\\", \\\"[\\\", RowBox[{StyleBox[\\\"eqns\\\", \ \\\"TI\\\"], \\\",\\\", StyleBox[\\\"vars\\\", \\\"TI\\\"], \\\",\\\", \ StyleBox[\\\"elims\\\", \\\"TI\\\"]}], \\\"]\\\"}]\\) attempts to solve the \ equations for \\!\\(\\*StyleBox[\\\"vars\\\", \\\"TI\\\"]\\), eliminating the \ variables \\!\\(\\*StyleBox[\\\"elims\\\", \\\"TI\\\"]\\). \"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Solve"]}]], "Print", "PrintUsage", CellChangeTimes->{3.4022248564844007`*^9}, CellTags->"Info3402199655-8263288"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "MatrixExp"}]], "Input", CellChangeTimes->{{3.402224840190175*^9, 3.4022248679238453`*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"MatrixExp\\\", \\\"[\\\", StyleBox[\\\"m\ \\\", \\\"TI\\\"], \\\"]\\\"}]\\) gives the matrix exponential of \ \\!\\(\\*StyleBox[\\\"m\\\", \\\"TI\\\"]\\). \ \\n\\!\\(\\*RowBox[{\\\"MatrixExp\\\", \\\"[\\\", RowBox[{StyleBox[\\\"m\\\", \ \\\"TI\\\"], \\\",\\\", StyleBox[\\\"v\\\", \\\"TI\\\"]}], \\\"]\\\"}]\\) \ gives the matrix exponential of \\!\\(\\*StyleBox[\\\"m\\\", \\\"TI\\\"]\\) \ applied to the vector \\!\\(\\*StyleBox[\\\"v\\\", \\\"TI\\\"]\\).\"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/MatrixExp"]}]], "Print", "PrintUsage", CellChangeTimes->{3.402224868965383*^9}, CellTags->"Info3402199668-6831718"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["4 x 4 example", "Subsection", CellChangeTimes->{{3.402126793823285*^9, 3.40212691650108*^9}, { 3.402127067991959*^9, 3.402127191093029*^9}, {3.4021272489713287`*^9, 3.402127271777747*^9}}, FontSize->12], Cell[TextData[{ "Note that in ", StyleBox["Mathematica", FontSlant->"Italic"], " matrices are entered as a list of rows (#@$!)." }], "Text", CellChangeTimes->{{3.402129233472764*^9, 3.402129259349321*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"a4a", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "1", ",", "3", ",", RowBox[{"-", "1"}]}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "0", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "1", ",", "2", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "1"}], ",", RowBox[{"-", "3"}], ",", "0"}], "}"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.402083276821743*^9, 3.4020833049485617`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigensystem", "[", "a4a", "]"}]], "Input", CellChangeTimes->{{3.4020833075934057`*^9, 3.402083310693364*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", RowBox[{"-", "1"}], ",", "1", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "5"}], ",", "3", ",", RowBox[{"-", "1"}], ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0"}], "}"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4021291007841883`*^9}] }, Open ]], Cell["\<\ Each eigenvector has only a one dimensional eigenspace. We need to find a \ preimage under s(A, \[Lambda]) of each basis vector:\ \>", "Text", CellChangeTimes->{{3.402129108258353*^9, 3.40212916463626*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"LinearSolve", "[", RowBox[{ RowBox[{"s", "[", RowBox[{"a4a", ",", RowBox[{"-", "1"}]}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "5"}], ",", "3", ",", RowBox[{"-", "1"}], ",", "5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.402083122529235*^9, 3.4020831240120077`*^9}, { 3.40208316629156*^9, 3.402083217180566*^9}, {3.402083329359212*^9, 3.402083342600623*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"2", ",", RowBox[{"-", "3"}], ",", "0", ",", "0"}], "}"}]], "Output", CellChangeTimes->{3.402083343566997*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"LinearSolve", "[", RowBox[{ RowBox[{"s", "[", RowBox[{"a4a", ",", "1"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.402083122529235*^9, 3.4020831240120077`*^9}, { 3.40208316629156*^9, 3.402083217180566*^9}, {3.402083329359212*^9, 3.402083360416275*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "1"}], ",", "0", ",", "0"}], "}"}]], "Output", CellChangeTimes->{3.402083360955228*^9}] }, Open ]], Cell["\<\ Now build the matrix U, following each eigenvector by its preimage. Since we \ want our vectors to be the columns, not the rows, of U, we need to transpose \ the matrix:\ \>", "Text", CellChangeTimes->{{3.4021291728943777`*^9, 3.402129220333467*^9}, { 3.40212926750974*^9, 3.402129291729546*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"u4a", "=", RowBox[{"Transpose", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "5"}], ",", "3", ",", RowBox[{"-", "1"}], ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", RowBox[{"-", "3"}], ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "1"}], ",", "0", ",", "0"}], "}"}]}], "}"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.402083387588338*^9, 3.402083418079144*^9}}], Cell[TextData[{ StyleBox["Verify ", FontSlant->"Italic"], "that we get the correct Jordan normal form. (If we didn't make any \ mistakes, we knew a priori that we would get this.)" }], "Text", CellChangeTimes->{{3.4021293031916447`*^9, 3.402129340045768*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Inverse", "[", "u4a", "]"}], ".", "a4a", ".", "u4a"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.402083420694919*^9, 3.402083437118767*^9}, { 3.402129440648793*^9, 3.40212944178745*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"-", "1"}], "1", "0", "0"}, {"0", RowBox[{"-", "1"}], "0", "0"}, {"0", "0", "1", "1"}, {"0", "0", "0", "1"} }, 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->{{3.402083429309054*^9, 3.402083437693554*^9}, { 3.402129403953226*^9, 3.40212944239625*^9}, 3.4021294822714243`*^9, 3.4021295361973133`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"MatrixExp", "[", RowBox[{"t", " ", "%"}], "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.402129394013954*^9, 3.4021294465080976`*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", "t"}], "0", "0"}, {"0", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], "0", "0"}, {"0", "0", SuperscriptBox["\[ExponentialE]", "t"], RowBox[{ SuperscriptBox["\[ExponentialE]", "t"], " ", "t"}]}, {"0", "0", "0", SuperscriptBox["\[ExponentialE]", "t"]} }, 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->{{3.402129399414303*^9, 3.402129447109374*^9}, 3.4021294842285147`*^9, 3.4021295381631804`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"u4a", ".", "%", ".", RowBox[{"Inverse", "[", "u4a", "]"}]}], "//", "Factor"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.40212945724441*^9, 3.402129488216997*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{"2", "-", RowBox[{"5", " ", "t"}], "+", RowBox[{"3", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], " ", "t"}]}], ")"}]}], RowBox[{ SuperscriptBox["\[ExponentialE]", "t"], " ", "t"}], RowBox[{ FractionBox["1", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "5"}], "+", RowBox[{"5", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]]}], "+", RowBox[{"2", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], " ", "t"}]}], ")"}]}], RowBox[{ FractionBox["1", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], "-", RowBox[{"10", " ", "t"}], "+", RowBox[{"4", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], " ", "t"}]}], ")"}]}]}, { RowBox[{ RowBox[{"-", FractionBox["3", "2"]}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", "t"]}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["\[ExponentialE]", "t"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "t"}], ")"}]}], RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", "t"]}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "t"}], ")"}]}], RowBox[{ RowBox[{"-", FractionBox["1", "4"]}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", RowBox[{"3", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]]}], "+", RowBox[{"2", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], " ", "t"}]}], ")"}]}], RowBox[{ RowBox[{"-", FractionBox["1", "4"]}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{"3", "-", RowBox[{"3", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]]}], "-", RowBox[{"6", " ", "t"}], "+", RowBox[{"4", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], " ", "t"}]}], ")"}]}]}, { RowBox[{ FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"3", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]]}]}], ")"}], " ", "t"}], RowBox[{ SuperscriptBox["\[ExponentialE]", "t"], " ", "t"}], RowBox[{ FractionBox["1", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"5", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]]}], "+", RowBox[{"2", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], " ", "t"}]}], ")"}]}], RowBox[{ FractionBox["1", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], "-", RowBox[{"2", " ", "t"}], "+", RowBox[{"4", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], " ", "t"}]}], ")"}]}]}, { RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "5"}], "+", RowBox[{"3", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]]}]}], ")"}], " ", "t"}], RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", "t"]}], " ", "t"}], RowBox[{ RowBox[{"-", FractionBox["1", "4"]}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "5"}], "+", RowBox[{"5", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]]}], "+", RowBox[{"2", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], " ", "t"}]}], ")"}]}], RowBox[{ RowBox[{"-", FractionBox["1", "4"]}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "5"}], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], "-", RowBox[{"10", " ", "t"}], "+", RowBox[{"4", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "t"}]], " ", "t"}]}], ")"}]}]} }, 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->{{3.402129472774618*^9, 3.402129489008803*^9}, 3.402129547989334*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"%", "-", RowBox[{"MatrixExp", "[", RowBox[{"t", " ", "a4a"}], "]"}]}], "//", "Factor"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.402129494153482*^9, 3.402129505563964*^9}, { 3.402129551130205*^9, 3.402129556818925*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0"}, {"0", "0", "0", "0"}, {"0", "0", "0", "0"}, {"0", "0", "0", "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->{3.4021295061822042`*^9, 3.4021295584083023`*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Another 4 x 4 example", "Subsection", CellChangeTimes->{{3.402126793823285*^9, 3.40212691650108*^9}, { 3.402127067991959*^9, 3.402127191093029*^9}, {3.4021272489713287`*^9, 3.402127271777747*^9}, {3.402129038033787*^9, 3.402129039281844*^9}}, FontSize->12], Cell[BoxData[ RowBox[{ RowBox[{"a4b", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "96"}], ",", "325", ",", "962", ",", RowBox[{"-", "1174"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "6", ",", "10", ",", RowBox[{"-", "12"}]}], "}"}], ",", RowBox[{"{", RowBox[{"6", ",", RowBox[{"-", "20"}], ",", RowBox[{"-", "55"}], ",", "71"}], "}"}], ",", RowBox[{"{", RowBox[{"13", ",", RowBox[{"-", "43"}], ",", RowBox[{"-", "126"}], ",", "157"}], "}"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.402084007014097*^9, 3.402084010397997*^9}, { 3.402127315630916*^9, 3.402127318347567*^9}, {3.40212761000788*^9, 3.402127613197954*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigensystem", "[", "a4b", "]"}]], "Input", CellChangeTimes->{{3.402083909488989*^9, 3.4020839114307613`*^9}, { 3.402084016779894*^9, 3.40208401797106*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"3", ",", "3", ",", "3", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "27"}], ",", RowBox[{"-", "1"}], ",", "0", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"13", ",", "1", ",", "1", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0"}], "}"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.402083912031283*^9, 3.402084018347249*^9}] }, Open ]], Cell["\<\ The sole eigenvalue of A is 3, with eigenspace span{(-27, -1, 0, 2), (13, 1, \ 1, 0)}.\ \>", "Text", CellChangeTimes->{{3.402127370996799*^9, 3.4021274447687063`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"s", "[", RowBox[{"a4b", ",", "3"}], "]"}]], "Input", CellChangeTimes->{3.402127598604762*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "99"}], ",", "325", ",", "962", ",", RowBox[{"-", "1174"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "3", ",", "10", ",", RowBox[{"-", "12"}]}], "}"}], ",", RowBox[{"{", RowBox[{"6", ",", RowBox[{"-", "20"}], ",", RowBox[{"-", "58"}], ",", "71"}], "}"}], ",", RowBox[{"{", RowBox[{"13", ",", RowBox[{"-", "43"}], ",", RowBox[{"-", "126"}], ",", "154"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4021275990778522`*^9, 3.402157913999114*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"LinearSolve", "[", RowBox[{ RowBox[{"s", "[", RowBox[{"a4b", ",", "3"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "27"}], ",", RowBox[{"-", "1"}], ",", "0", ",", "2"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.402084029635016*^9, 3.402084051798867*^9}, { 3.4020841583743134`*^9, 3.4020842003877163`*^9}, {3.402084524560186*^9, 3.4020845491947947`*^9}, {3.402127501736786*^9, 3.402127509220141*^9}, { 3.40215901159295*^9, 3.4021590238431892`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"LinearSolve", "::", "\<\"nosol\"\>"}], RowBox[{ ":", " "}], "\<\"Linear equation encountered that has no solution. \ \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/LinearSolve/nosol\\\", \ ButtonNote -> \\\"LinearSolve::nosol\\\"]\\)\"\>"}]], "Message", "MSG", CellChangeTimes->{3.402159024664353*^9}, FontSize->12], Cell[BoxData[ RowBox[{"LinearSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "99"}], ",", "325", ",", "962", ",", RowBox[{"-", "1174"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "3", ",", "10", ",", RowBox[{"-", "12"}]}], "}"}], ",", RowBox[{"{", RowBox[{"6", ",", RowBox[{"-", "20"}], ",", RowBox[{"-", "58"}], ",", "71"}], "}"}], ",", RowBox[{"{", RowBox[{"13", ",", RowBox[{"-", "43"}], ",", RowBox[{"-", "126"}], ",", "154"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "27"}], ",", RowBox[{"-", "1"}], ",", "0", ",", "2"}], "}"}]}], "]"}]], "Output", CellChangeTimes->{3.4021590247092943`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"LinearSolve", "[", RowBox[{ RowBox[{"s", "[", RowBox[{"a4b", ",", "3"}], "]"}], ",", RowBox[{"{", RowBox[{"13", ",", "1", ",", "1", ",", "0"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.402084029635016*^9, 3.402084051798867*^9}, { 3.4020841583743134`*^9, 3.4020842003877163`*^9}, {3.402084524560186*^9, 3.4020845491947947`*^9}, {3.402127501736786*^9, 3.402127509220141*^9}, { 3.40215901159295*^9, 3.4021590238431892`*^9}, {3.402224908316965*^9, 3.402224912530442*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"LinearSolve", "::", "\<\"nosol\"\>"}], RowBox[{ ":", " "}], "\<\"Linear equation encountered that has no solution. \ \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/LinearSolve/nosol\\\", \ ButtonNote -> \\\"LinearSolve::nosol\\\"]\\)\"\>"}]], "Message", "MSG", CellChangeTimes->{3.402224913247302*^9}], Cell[BoxData[ RowBox[{"LinearSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "99"}], ",", "325", ",", "962", ",", RowBox[{"-", "1174"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "3", ",", "10", ",", RowBox[{"-", "12"}]}], "}"}], ",", RowBox[{"{", RowBox[{"6", ",", RowBox[{"-", "20"}], ",", RowBox[{"-", "58"}], ",", "71"}], "}"}], ",", RowBox[{"{", RowBox[{"13", ",", RowBox[{"-", "43"}], ",", RowBox[{"-", "126"}], ",", "154"}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{"13", ",", "1", ",", "1", ",", "0"}], "}"}]}], "]"}]], "Output", CellChangeTimes->{3.4022249132720013`*^9}] }, Open ]], Cell["\<\ We need to get hold of a vector in the eigenspace that' s also in the range \ of s(A, 3), and hence has a preimage:\ \>", "Text", CellChangeTimes->{{3.4022249167247868`*^9, 3.402224979953524*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "[", RowBox[{"a4b", ",", "3"}], "]"}], ".", RowBox[{"{", RowBox[{"w1", ",", "w2", ",", "w3", ",", "w4"}], "}"}]}], "\[Equal]", RowBox[{ RowBox[{"v1", RowBox[{"{", RowBox[{ RowBox[{"-", "27"}], ",", RowBox[{"-", "1"}], ",", "0", ",", "2"}], "}"}]}], "+", RowBox[{"v2", RowBox[{"{", RowBox[{"13", ",", "1", ",", "1", ",", "0"}], "}"}]}]}]}], ",", RowBox[{"{", RowBox[{"w1", ",", "w2", ",", "w3", ",", "w4", ",", "v1", ",", "v2"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.402084029635016*^9, 3.402084051798867*^9}, { 3.4020841583743134`*^9, 3.4020842003877163`*^9}, {3.402084524560186*^9, 3.4020845491947947`*^9}, {3.402127501736786*^9, 3.402127509220141*^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->{3.4020841615983973`*^9, 3.402084201780098*^9, 3.402084365274742*^9, 3.402084403453356*^9, 3.402084496870657*^9, 3.4020845515132923`*^9, 3.4020846276713667`*^9, 3.40212754021555*^9, 3.402127586782936*^9, 3.4021276180786123`*^9}, FontSize->12], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"w1", "\[Rule]", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"3", " ", "v2"}], "2"]}], "+", RowBox[{"13", " ", "w3"}], "-", FractionBox[ RowBox[{"27", " ", "w4"}], "2"]}]}], ",", RowBox[{"w2", "\[Rule]", RowBox[{ RowBox[{"-", FractionBox["v2", "2"]}], "+", "w3", "-", FractionBox["w4", "2"]}]}], ",", RowBox[{"v1", "\[Rule]", "v2"}]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.402084052328823*^9, 3.402084161679575*^9, 3.402084201880926*^9, 3.402084365424073*^9, 3.4020844035649967`*^9, 3.402084496892172*^9, 3.402084551623516*^9, 3.4020846277837343`*^9, 3.4021275405591288`*^9, 3.402127586926293*^9, 3.402127618100174*^9}] }, Open ]], Cell["\<\ Note that the condition v1 = v2 implies that only a one dimensional subspace, \ span{(-27, -1, 0, 2) + (13, 1, 1, 0)} = span{(-14, 0, 1, 2)}, of the eigenspace intersects range(s(A, 3)), i.e. has a preimage under s (A, \ 3).\ \>", "Text", CellChangeTimes->{{3.402127661034683*^9, 3.4021278102072487`*^9}, { 3.402224994020732*^9, 3.402225002068469*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"w1", ",", "w2", ",", "w3", ",", "w4"}], "}"}], "/.", RowBox[{ "%", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}]], "Input", CellChangeTimes->{{3.402084309674674*^9, 3.402084331684392*^9}, { 3.402084370321726*^9, 3.402084372552292*^9}, {3.40208440532628*^9, 3.402084407262504*^9}, {3.402084489321516*^9, 3.402084494915497*^9}, { 3.40208456134809*^9, 3.402084564954699*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"3", " ", "v2"}], "2"]}], "+", RowBox[{"13", " ", "w3"}], "-", FractionBox[ RowBox[{"27", " ", "w4"}], "2"]}], ",", RowBox[{ RowBox[{"-", FractionBox["v2", "2"]}], "+", "w3", "-", FractionBox["w4", "2"]}], ",", "w3", ",", "w4"}], "}"}]], "Output", CellChangeTimes->{3.402084333077302*^9, 3.402084375748168*^9, 3.402084407949212*^9, 3.402084499803043*^9, 3.40208456600392*^9, 3.4020846300044003`*^9, 3.402127636703367*^9}] }, Open ]], Cell["\<\ We need to do one more \"bounce\", to find a preimage of this vector:\ \>", "Text", CellChangeTimes->{{3.402127830320552*^9, 3.4021278839832163`*^9}, { 3.402225012680822*^9, 3.402225031839768*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"s", "[", RowBox[{"a4b", ",", "3"}], "]"}], ".", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3", ",", "x4"}], "}"}]}], "\[Equal]", "%"}], ",", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3", ",", "x4", ",", "w3", ",", "w4"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.402084029635016*^9, 3.402084051798867*^9}, { 3.4020841583743134`*^9, 3.4020842003877163`*^9}, {3.402084524560186*^9, 3.402084532030076*^9}, {3.402084570215423*^9, 3.4020846328252974`*^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->{{3.402084591033154*^9, 3.402084610363235*^9}, 3.402084655359209*^9}, FontSize->12], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"x1", "\[Rule]", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"3", " ", "w4"}], "4"]}], "+", RowBox[{"13", " ", "x3"}], "-", FractionBox[ RowBox[{"27", " ", "x4"}], "2"], "+", FractionBox[ RowBox[{"43", " ", "y2"}], "4"]}]}], ",", RowBox[{"x2", "\[Rule]", RowBox[{ RowBox[{"-", FractionBox["w4", "4"]}], "+", "x3", "-", FractionBox["x4", "2"], "+", FractionBox[ RowBox[{"13", " ", "y2"}], "4"]}]}], ",", RowBox[{"w3", "\[Rule]", RowBox[{ FractionBox["w4", "2"], "-", FractionBox["y2", "2"]}]}]}], "}"}], "}"}]], "Output", CellChangeTimes->{{3.402084591267652*^9, 3.402084610476467*^9}, 3.4020846554665117`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"%%", ",", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3", ",", "x4"}], "}"}]}], "}"}], "/.", RowBox[{ "%", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}]], "Input", CellChangeTimes->{{3.402084309674674*^9, 3.402084331684392*^9}, { 3.402084370321726*^9, 3.402084372552292*^9}, {3.40208440532628*^9, 3.402084407262504*^9}, {3.402084489321516*^9, 3.402084494915497*^9}, { 3.4020846684441223`*^9, 3.402084676973626*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"27", " ", "w4"}], "2"]}], "+", RowBox[{"13", " ", RowBox[{"(", RowBox[{ FractionBox["w4", "2"], "-", FractionBox["y2", "2"]}], ")"}]}], "-", FractionBox[ RowBox[{"3", " ", "y2"}], "2"]}], ",", RowBox[{"-", "y2"}], ",", RowBox[{ FractionBox["w4", "2"], "-", FractionBox["y2", "2"]}], ",", "w4"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"3", " ", "w4"}], "4"]}], "+", RowBox[{"13", " ", "x3"}], "-", FractionBox[ RowBox[{"27", " ", "x4"}], "2"], "+", FractionBox[ RowBox[{"43", " ", "y2"}], "4"]}], ",", RowBox[{ RowBox[{"-", FractionBox["w4", "4"]}], "+", "x3", "-", FractionBox["x4", "2"], "+", FractionBox[ RowBox[{"13", " ", "y2"}], "4"]}], ",", "x3", ",", "x4"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.402084333077302*^9, 3.402084375748168*^9, 3.402084407949212*^9, 3.402084499803043*^9, 3.4020846779002943`*^9}] }, Open ]], Cell["\<\ We can now choose values for the free parameters y2, w4, x3, and x4; y2 needs \ to be nonzero, or our original element of the eigenspace will be the zero \ vector!\ \>", "Text", CellChangeTimes->{{3.402127900831477*^9, 3.402127966611669*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"%", "/.", RowBox[{"{", RowBox[{ RowBox[{"y2", "\[Rule]", "1"}], ",", RowBox[{"w4", "\[Rule]", "0"}], ",", RowBox[{"x3", "\[Rule]", "0"}], ",", RowBox[{"x4", "\[Rule]", "0"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.4020847073368607`*^9, 3.402084723529532*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "8"}], ",", RowBox[{"-", "1"}], ",", RowBox[{"-", FractionBox["1", "2"]}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ FractionBox["43", "4"], ",", FractionBox["13", "4"], ",", "0", ",", "0"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4020847240815287`*^9}] }, Open ]], Cell["The corresponding matrix U is :", "Text", CellChangeTimes->{{3.40212801638583*^9, 3.402128023843603*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"u4b", "=", RowBox[{"Transpose", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"13", ",", "1", ",", "1", ",", "0"}], "}"}], ",", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "27"}], ",", RowBox[{"-", "1"}], ",", "0", ",", "2"}], "}"}], "+", RowBox[{"{", RowBox[{"13", ",", "1", ",", "1", ",", "0"}], "}"}]}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "8"}], ",", RowBox[{"-", "1"}], ",", RowBox[{"-", FractionBox["1", "2"]}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ FractionBox["43", "4"], ",", FractionBox["13", "4"], ",", "0", ",", "0"}], "}"}]}], "}"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.402084736494849*^9, 3.402084776551218*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Inverse", "[", "u4b", "]"}], ".", "a4b", ".", "u4b"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.40208410596266*^9, 3.4020841119540443`*^9}, { 3.402084792328813*^9, 3.402084793988841*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"3", "0", "0", "0"}, {"0", "3", "1", "0"}, {"0", "0", "3", "1"}, {"0", "0", "0", "3"} }, 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->{ 3.402084113343883*^9, {3.402084788799526*^9, 3.402084794545566*^9}, 3.402084903857109*^9, 3.4020849822611513`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"s", "[", RowBox[{"%", ",", "3"}], "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.4020848453792753`*^9, 3.402084851471821*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0"}, {"0", "0", "1", "0"}, {"0", "0", "0", "1"}, {"0", "0", "0", "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->{3.402084852174748*^9, 3.402084905870141*^9, 3.4020849845118504`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"IdentityMatrix", "[", "4", "]"}], "+", RowBox[{"t", "%"}], "+", RowBox[{ FractionBox[ SuperscriptBox["t", "2"], "2"], RowBox[{"%", ".", "%"}]}]}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.402084856118469*^9, 3.402084868968449*^9}, 3.402084920470401*^9}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0"}, {"0", "1", "t", FractionBox[ SuperscriptBox["t", "2"], "2"]}, {"0", "0", "1", "t"}, {"0", "0", "0", "1"} }, 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->{3.4020848760977573`*^9, 3.402084921065319*^9, 3.4020849870858393`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["E", RowBox[{"3", "t"}]], RowBox[{"u4b", ".", "%", ".", RowBox[{"Inverse", "[", "u4b", "]"}]}]}], "//", "Factor"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.402084884789462*^9, 3.402084928868452*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]]}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"99", " ", "t"}], "+", RowBox[{"7", " ", SuperscriptBox["t", "2"]}]}], ")"}]}], RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", "t", " ", RowBox[{"(", RowBox[{"325", "+", RowBox[{"21", " ", "t"}]}], ")"}]}], RowBox[{"2", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", "t", " ", RowBox[{"(", RowBox[{"481", "+", RowBox[{"35", " ", "t"}]}], ")"}]}], RowBox[{ RowBox[{"-", "2"}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", "t", " ", RowBox[{"(", RowBox[{"587", "+", RowBox[{"42", " ", "t"}]}], ")"}]}]}, { RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]]}], " ", "t"}], RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"3", " ", "t"}]}], ")"}]}], RowBox[{"10", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", "t"}], RowBox[{ RowBox[{"-", "12"}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", "t"}]}, { RowBox[{ FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", "t", " ", RowBox[{"(", RowBox[{"12", "+", "t"}], ")"}]}], RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", "t", " ", RowBox[{"(", RowBox[{"40", "+", RowBox[{"3", " ", "t"}]}], ")"}]}], RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]]}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"58", " ", "t"}], "+", RowBox[{"5", " ", SuperscriptBox["t", "2"]}]}], ")"}]}], RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", "t", " ", RowBox[{"(", RowBox[{"71", "+", RowBox[{"6", " ", "t"}]}], ")"}]}]}, { RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", "t", " ", RowBox[{"(", RowBox[{"13", "+", "t"}], ")"}]}], RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]]}], " ", "t", " ", RowBox[{"(", RowBox[{"43", "+", RowBox[{"3", " ", "t"}]}], ")"}]}], RowBox[{ RowBox[{"-", "2"}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", "t", " ", RowBox[{"(", RowBox[{"63", "+", RowBox[{"5", " ", "t"}]}], ")"}]}], RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"3", " ", "t"}]], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"154", " ", "t"}], "+", RowBox[{"12", " ", SuperscriptBox["t", "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->{3.402084897361397*^9, 3.402084929535717*^9, 3.402084990642754*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"%", "-", RowBox[{"MatrixExp", "[", RowBox[{"t", " ", "a4b"}], "]"}]}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.402084933526026*^9, 3.402084946731831*^9}, { 3.402084993390195*^9, 3.40208499370219*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0"}, {"0", "0", "0", "0"}, {"0", "0", "0", "0"}, {"0", "0", "0", "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->{3.402084947370743*^9, 3.40208499668441*^9}] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{713, 521}, WindowMargins->{{8, Automatic}, {Automatic, 0}}, ShowSelection->True, FrontEndVersion->"6.0 for Mac OS X PowerPC (32-bit) (April 20, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{ "Info3402199648-9239799"->{ Cell[2064, 66, 1026, 19, 55, "Print", CellTags->"Info3402199648-9239799"]}, "Info3402199655-8263288"->{ Cell[3246, 94, 941, 18, 55, "Print", CellTags->"Info3402199655-8263288"]}, "Info3402199668-6831718"->{ Cell[4347, 121, 778, 16, 55, "Print", CellTags->"Info3402199668-6831718"]} } *) (*CellTagsIndex CellTagsIndex->{ {"Info3402199648-9239799", 45458, 1412}, {"Info3402199655-8263288", 45568, 1415}, {"Info3402199668-6831718", 45677, 1418} } *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 317, 6, 52, "Title"], Cell[CellGroupData[{ Cell[932, 33, 258, 4, 32, "Subsection"], Cell[1193, 39, 312, 8, 27, "Input"], Cell[1508, 49, 406, 9, 26, "Text"], Cell[CellGroupData[{ Cell[1939, 62, 122, 2, 27, "Input"], Cell[2064, 66, 1026, 19, 55, "Print", CellTags->"Info3402199648-9239799"] }, Open ]], Cell[CellGroupData[{ Cell[3127, 90, 116, 2, 27, "Input"], Cell[3246, 94, 941, 18, 55, "Print", CellTags->"Info3402199655-8263288"] }, Open ]], Cell[CellGroupData[{ Cell[4224, 117, 120, 2, 27, "Input"], Cell[4347, 121, 778, 16, 55, "Print", CellTags->"Info3402199668-6831718"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[5174, 143, 215, 4, 32, "Subsection"], Cell[5392, 149, 211, 6, 26, "Text"], Cell[5606, 157, 613, 18, 27, "Input"], Cell[CellGroupData[{ Cell[6244, 179, 134, 2, 27, "Input"], Cell[6381, 183, 684, 22, 27, "Output"] }, Open ]], Cell[7080, 208, 217, 4, 26, "Text"], Cell[CellGroupData[{ Cell[7322, 216, 440, 12, 27, "Input"], Cell[7765, 230, 154, 4, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7956, 239, 419, 11, 27, "Input"], Cell[8378, 252, 154, 4, 27, "Output"] }, Open ]], Cell[8547, 259, 309, 6, 41, "Text"], Cell[8859, 267, 690, 21, 27, "Input"], Cell[9552, 290, 264, 6, 26, "Text"], Cell[CellGroupData[{ Cell[9841, 300, 250, 6, 27, "Input"], Cell[10094, 308, 868, 24, 89, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10999, 337, 187, 4, 27, "Input"], Cell[11189, 343, 1153, 32, 101, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[12379, 380, 230, 6, 27, "Input"], Cell[12612, 388, 6394, 188, 125, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19043, 581, 296, 8, 27, "Input"], Cell[19342, 591, 725, 20, 89, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[20116, 617, 269, 4, 32, "Subsection"], Cell[20388, 623, 796, 23, 43, "Input"], Cell[CellGroupData[{ Cell[21209, 650, 182, 3, 27, "Input"], Cell[21394, 655, 618, 18, 27, "Output"] }, Open ]], Cell[22027, 676, 178, 4, 26, "Text"], Cell[CellGroupData[{ Cell[22230, 684, 121, 3, 27, "Input"], Cell[22354, 689, 622, 19, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[23013, 713, 521, 12, 27, "Input"], Cell[23537, 727, 438, 9, 23, "Message"], Cell[23978, 738, 808, 25, 43, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[24823, 768, 528, 11, 27, "Input"], Cell[25354, 781, 423, 8, 23, "Message"], Cell[25780, 791, 766, 23, 43, "Output"] }, Open ]], Cell[26561, 817, 207, 4, 26, "Text"], Cell[CellGroupData[{ Cell[26793, 825, 859, 23, 43, "Input"], Cell[27655, 850, 650, 12, 23, "Message"], Cell[28308, 864, 791, 21, 44, "Output"] }, Open ]], Cell[29114, 888, 367, 8, 56, "Text"], Cell[CellGroupData[{ Cell[29506, 900, 448, 9, 27, "Input"], Cell[29957, 911, 564, 16, 44, "Output"] }, Open ]], Cell[30536, 930, 210, 4, 26, "Text"], Cell[CellGroupData[{ Cell[30771, 938, 595, 15, 27, "Input"], Cell[31369, 955, 486, 10, 23, "Message"], Cell[31858, 967, 815, 26, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[32710, 998, 503, 11, 27, "Input"], Cell[33216, 1011, 1204, 39, 79, "Output"] }, Open ]], Cell[34435, 1053, 253, 5, 26, "Text"], Cell[CellGroupData[{ Cell[34713, 1062, 313, 8, 27, "Input"], Cell[35029, 1072, 393, 13, 44, "Output"] }, Open ]], Cell[35437, 1088, 112, 1, 26, "Text"], Cell[35552, 1091, 861, 26, 80, "Input"], Cell[CellGroupData[{ Cell[36438, 1121, 252, 6, 27, "Input"], Cell[36693, 1129, 798, 22, 89, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[37528, 1156, 179, 4, 27, "Input"], Cell[37710, 1162, 748, 21, 89, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[38495, 1188, 339, 10, 48, "Input"], Cell[38837, 1200, 804, 23, 101, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[39678, 1228, 299, 9, 30, "Input"], Cell[39980, 1239, 4094, 124, 111, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[44111, 1368, 260, 6, 27, "Input"], Cell[44374, 1376, 720, 20, 89, "Output"] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)