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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[ 39812, 1084]*) (*NotebookOutlinePosition[ 40969, 1121]*) (* CellTagsIndexPosition[ 40925, 1117]*) (*WindowFrame->Normal*) Notebook[{ Cell["Generalized eigenvalue problems", "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This notebook has been written in ", StyleBox["Mathematica ", FontSlant->"Italic"], "by \n\n", StyleBox["Mark J. McCready\nProfessor and Chair of Chemical Engineering\n\ University of Notre Dame\nNotre Dame IN 46556\nUSA", FontSize->14], "\n\nMark.J.McCready.1@nd.edu\n", ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \ the World Wide Web and the Internet.\n\nI would hope that as a professional \ courtesy, this notice remain visible to other users. \nThere is no charge for \ copying and dissemination \n\nVersion: 10/6/98\nMore recent versions of \ this notebook should be available at the web site:\n", Cell[BoxData[ FormBox[ ButtonBox[\(\(\(http\)\(:\)\) // \(www . nd . edu/\(\(~\)\(mjm\)\)\)/ generalized . eigenvalues . nb\), ButtonData:>{ URL[ "http://www.nd.edu/~mjm/generalized.eigenvalues.nb"], None}, ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text"], Cell[TextData[{ "For a problem where ", Cell[BoxData[ \(TraditionalForm\`\(AB(\[Lambda])\)\ y\ = \ 0\)]], ", we expect that non trivial solutions for y will exist only for certain \ values of \[Lambda]. Thus this problem appears to be an eigenvalue problem, \ but not of the usual form. ", Cell[BoxData[ \(TraditionalForm\`\((A\ - \ I\ \[Lambda])\)\ x\ = 0\)]], ". Can we convert ", Cell[BoxData[ \(TraditionalForm\`\(AB(\[Lambda])\)\ y\ = \ 0\)]], " to the standard form? Yes, we realize a \"generalized\" version of ", Cell[BoxData[ \(TraditionalForm\`\(\(AB(\[Lambda])\)\ y\ = \)\)]], "0 is ", Cell[BoxData[ \(TraditionalForm\`Ax = B\[Lambda]\ x\)]], ". \n" }], "Text"], Cell[CellGroupData[{ Cell["First try a simple arbitrary matrix", "Subsection"], Cell[CellGroupData[{ Cell["Here is the AB matrix", "Text"], Cell[CellGroupData[{ Cell[TextData[ "AB={{1-\[Lambda],2,-3+\[Lambda]},{2 \[Lambda], -3, 4+\[Lambda]},{11,-3 \ \[Lambda]+3, 2-\[Lambda]}} "], "Input", CellLabel->"In[9]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {\(1 - \[Lambda]\), "2", \(\[Lambda] - 3\)}, {\(2\ \[Lambda]\), \(-3\), \(\[Lambda] + 4\)}, {"11", \(3 - 3\ \[Lambda]\), \(2 - \[Lambda]\)} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[9]="] }, Open ]], Cell[TextData["We can get A by simply removing \[Lambda]. "], "Text"], Cell[CellGroupData[{ Cell[TextData["A=AB/.\[Lambda]->0"], "Input", CellLabel->"In[10]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"1", "2", \(-3\)}, {"0", \(-3\), "4"}, {"11", "3", "2"} }, ColumnAlignments->{Decimal}], ")"}], TraditionalForm]], "Output", CellLabel->"Out[10]="] }, Open ]], Cell[TextData[ "Now looking at the definition we can get B by subtracting A from AB and \ factoring out \[Lambda]. "], "Text"], Cell[CellGroupData[{ Cell[TextData["B=-(AB-A)/\[Lambda]"], "Input", CellLabel->"In[11]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"1", "0", \(-1\)}, {\(-2\), "0", \(-1\)}, {"0", "3", "1"} }, ColumnAlignments->{Decimal}], ")"}], TraditionalForm]], "Output", CellLabel->"Out[11]="] }, Open ]], Cell["NOw check to see what we have", "Text"], Cell[CellGroupData[{ Cell[TextData["A - B \[Lambda]"], "Input", CellLabel->"In[12]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {\(1 - \[Lambda]\), "2", \(\[Lambda] - 3\)}, {\(2\ \[Lambda]\), \(-3\), \(\[Lambda] + 4\)}, {"11", \(3 - 3\ \[Lambda]\), \(2 - \[Lambda]\)} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[12]="] }, Open ]], Cell["So it works. ", "Text"], Cell[TextData[{ "Now how do we solve, Ax = B \[Lambda] x?\n\nForm ", Cell[BoxData[ \(TraditionalForm \`B\^\(-1\)\ A\ x\ = \ B\^\(-1\)\ B\ \[Lambda]\ x\)]], ".\n\nThis gives:\n\n", Cell[BoxData[ \(TraditionalForm\`\(\(\((B\^\(-1\)\ A\ - I\ \[Lambda]\ )\) x\ = \ 0, \)\ \)\)]], "which is of the usual form of an eigen value problem.\n" }], "Text"], Cell[CellGroupData[{ Cell[TextData["temp1=Inverse[B].(A - B \[Lambda])"], "Input", CellLabel->"In[15]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {\(\(1 - \[Lambda]\)\/3 - \(2\ \[Lambda]\)\/3\), \(5\/3\), \(1\/3\ \((\(-\[Lambda]\) - 4)\) + \(\[Lambda] - 3\)\/3\)}, {\(\(2\ \((1 - \[Lambda])\)\)\/9 + \(2\ \[Lambda]\)\/9 + 11\/3\), \(1\/3\ \((3 - 3\ \[Lambda])\) + 1\/9\), \(\(2 - \[Lambda]\)\/3 + \(2\ \((\[Lambda] - 3)\)\)\/9 + \(\[Lambda] + 4\)\/9\)}, {\(\(-\(2\/3\)\)\ \((1 - \[Lambda])\) - \(2\ \[Lambda]\)\/3\), \(-\(1\/3\)\), \(1\/3\ \((\(-\[Lambda]\) - 4)\) - \(2\ \((\[Lambda] - 3)\)\)\/3\)} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[15]="] }, Open ]], Cell["We now have an eigen value problem", "Text"], Cell[CellGroupData[{ Cell["temp2=Simplify[temp1]", "Input", CellLabel->"In[16]:="], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {\(1\/3 - \[Lambda]\), \(5\/3\), \(-\(7\/3\)\)}, {\(35\/9\), \(10\/9 - \[Lambda]\), \(4\/9\)}, {\(-\(2\/3\)\), \(-\(1\/3\)\), \(2\/3 - \[Lambda]\)} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[16]="] }, Open ]], Cell[CellGroupData[{ Cell["temp3=Det[temp2]", "Input", CellLabel->"In[17]:="], Cell[BoxData[ \(TraditionalForm \`\(-\[Lambda]\^3\) + \(19\ \[Lambda]\^2\)\/9 + \(59\ \[Lambda]\)\/9 - 29\/9\)], "Output", CellLabel->"Out[17]="] }, Open ]], Cell[CellGroupData[{ Cell[TextData["eigs=Solve[%==0,\[Lambda]]"], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \(TraditionalForm \`{{\[Lambda] \[Rule] 19\/27 - \(\((1 + \[ImaginaryI]\ \@3)\)\ \@\(10274 + 81\ \[ImaginaryI]\ \@268190\)\%3\)\/\(27\ 2\^\(2/3\)\) - \(977\ \((1 - \[ImaginaryI]\ \@3)\)\)\/\(27\ \@\(2\ \((10274 + 81\ \[ImaginaryI]\ \@268190)\)\)\%3\)}, { \[Lambda] \[Rule] 19\/27 - \(\((1 - \[ImaginaryI]\ \@3)\)\ \@\(10274 + 81\ \[ImaginaryI]\ \@268190\)\%3\)\/\(27\ 2\^\(2/3\)\) - \(977\ \((1 + \[ImaginaryI]\ \@3)\)\)\/\(27\ \@\(2\ \((10274 + 81\ \[ImaginaryI]\ \@268190)\)\)\%3\)}, { \[Lambda] \[Rule] 19\/27 + \(977\ 2\^\(2/3\)\)\/\(27\ \@\(10274 + 81\ \[ImaginaryI]\ \@268190\)\%3\) + 1\/27\ \@\(2\ \((10274 + 81\ \[ImaginaryI]\ \@268190)\)\)\%3}} \)], "Output", CellLabel->"Out[18]="] }, Open ]], Cell[CellGroupData[{ Cell["Chop[N[eigs]]", "Input", CellLabel->"In[20]:="], Cell[BoxData[ \(TraditionalForm \`{{\[Lambda] \[Rule] 0.441819086396896842`}, { \[Lambda] \[Rule] \(-1.99196230836722616`\)}, { \[Lambda] \[Rule] 3.66125433308144021`}}\)], "Output", CellLabel->"Out[20]="] }, Open ]], Cell["Which is the same as", "Text"], Cell[CellGroupData[{ Cell["Chop[N[Eigenvalues[Inverse[B].A]]]", "Input", CellLabel->"In[23]:="], Cell[BoxData[ \(TraditionalForm \`{0.441819086396896842`, \(-1.99196230836722616`\), 3.66125433308144021`} \)], "Output", CellLabel->"Out[23]="] }, Open ]], Cell["We might also expect that less work is required:", "Text"], Cell[CellGroupData[{ Cell["Det[AB]", "Input", CellLabel->"In[24]:="], Cell[BoxData[ \(TraditionalForm \`\(-9\)\ \[Lambda]\^3 + 19\ \[Lambda]\^2 + 59\ \[Lambda] - 29\)], "Output", CellLabel->"Out[24]="] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Solve[%==0,\[Lambda]]"], "Input", CellLabel->"In[25]:="], Cell[BoxData[ \(TraditionalForm \`{{\[Lambda] \[Rule] 19\/27 - \(\((1 + \[ImaginaryI]\ \@3)\)\ \@\(10274 + 81\ \[ImaginaryI]\ \@268190\)\%3\)\/\(27\ 2\^\(2/3\)\) - \(977\ \((1 - \[ImaginaryI]\ \@3)\)\)\/\(27\ \@\(2\ \((10274 + 81\ \[ImaginaryI]\ \@268190)\)\)\%3\)}, { \[Lambda] \[Rule] 19\/27 - \(\((1 - \[ImaginaryI]\ \@3)\)\ \@\(10274 + 81\ \[ImaginaryI]\ \@268190\)\%3\)\/\(27\ 2\^\(2/3\)\) - \(977\ \((1 + \[ImaginaryI]\ \@3)\)\)\/\(27\ \@\(2\ \((10274 + 81\ \[ImaginaryI]\ \@268190)\)\)\%3\)}, { \[Lambda] \[Rule] 19\/27 + \(977\ 2\^\(2/3\)\)\/\(27\ \@\(10274 + 81\ \[ImaginaryI]\ \@268190\)\%3\) + 1\/27\ \@\(2\ \((10274 + 81\ \[ImaginaryI]\ \@268190)\)\)\%3}} \)], "Output", CellLabel->"Out[25]="] }, Open ]], Cell[CellGroupData[{ Cell["Chop[N[%]]", "Input", CellLabel->"In[26]:="], Cell[BoxData[ \(TraditionalForm \`{{\[Lambda] \[Rule] 0.441819086396896842`}, { \[Lambda] \[Rule] \(-1.99196230836722616`\)}, { \[Lambda] \[Rule] 3.66125433308144021`}}\)], "Output", CellLabel->"Out[26]="] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Now try a matrix from the numerical solution to channel flow \ stability\ \>", "Subsection"], Cell[TextData[{ "The complete motivation for this example can be found in the mathematica \ notebook,\n", Cell[BoxData[ FormBox[ ButtonBox[\(Linear\ Stability\ of\ Pressure\ Driven\ Channel\ Flow\), ButtonData:>{ URL[ "http://www.nd.edu/~mjm/channelflow.nb"], None}, ButtonStyle->"Hyperlink"], TraditionalForm]]], ". " }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox[ "The matrix that arises when solving the Orr-Sommerfeld equation using \ Chebyshev polynomials in a spectral method is shown below as AB. The \ physical problem is the stability of laminar channel flow to turbulence \ (although there is a problem with this) The term rr is the Reynolds number \ and \[Alpha] is the wave number. In a spectral method, the solution is \ expressed in terms of a set of orthogonal functions with unknown \ coefficients. Chebyshev polynomials are a convenient set for numerical \ problems on regular finite domains. They are solutions to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Chebyshev's", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" differential equation and the first few are", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["chebs=Table[ChebyshevT[i,y],{i,0,5}]", "Input", CellLabel->"In[37]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm \`{1, y, 2\ y\^2 - 1, 4\ y\^3 - 3\ y, 8\ y\^4 - 8\ y\^2 + 1, 16\ y\^5 - 20\ y\^3 + 5\ y}\)], "Output", CellLabel->"Out[37]="] }, Open ]], Cell[TextData[ "Here is the type of orthogonity that these display. You can think of this \ integration as an appropriate inner product for these functions on the \ domain, -1,1. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ Table[Integrate[chebs*ChebyshevT[i,y]* (1-y^2)^(-1/2),{y,-1,1}],{i,0,5}]\ \>", "Input", CellLabel->"In[38]:=", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"\[Pi]", "0", "0", "0", "0", "0"}, {"0", \(\[Pi]\/2\), "0", "0", "0", "0"}, {"0", "0", \(\[Pi]\/2\), "0", "0", "0"}, {"0", "0", "0", \(\[Pi]\/2\), "0", "0"}, {"0", "0", "0", "0", \(\[Pi]\/2\), "0"}, {"0", "0", "0", "0", "0", \(\[Pi]\/2\)} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[38]="] }, Open ]], Cell[TextData["The matrix is"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"AB", "=", RowBox[{"(", GridBox[{ { \(\[Alpha]\^4 + 1\/2\ \[ImaginaryI]\ rr\ \[Alpha]\^3 + \(10\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 576\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] - \(1152\ \[Alpha]\^2\)\/7 - \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha]\^3 + 271872\/7\), \(\(-\(1\/4\)\)\ \[ImaginaryI]\ rr\ \[Alpha]\^3 + \(12\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 456\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] - \(912\ \[Alpha]\^2\)\/7 + 238080\/7\), \(\(4\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 208\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] - \(416\ \[Alpha]\^2\)\/7 + 138048\/7\)}, { \(\(-\(1\/2\)\)\ \[ImaginaryI]\ rr\ \[Alpha]\^3 + \(48\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 1248\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] - \(2496\ \[Alpha]\^2\)\/7 + 460800\/7\), \(\[Alpha]\^4 + 1\/2\ \[ImaginaryI]\ rr\ \[Alpha]\^3 + \(52\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 960\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] - \(1920\ \[Alpha]\^2\)\/7 - \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha]\^3 + 405120\/7\), \(\(-\(1\/4\)\)\ \[ImaginaryI]\ rr\ \[Alpha]\^3 + \(8\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 432\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] - \(864\ \[Alpha]\^2\)\/7 + 238080\/7\)}, { \(4\/7\ \[ImaginaryI]\ rr\ \[Alpha]\^3 + \(48\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 1536\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] - \(3072\ \[Alpha]\^2\)\/7 + 34560\), \(2\/7\ \[ImaginaryI]\ rr\ \[Alpha]\^3 + \(52\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 1272\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] - \(2544\ \[Alpha]\^2\)\/7 + 30720\), \(\[Alpha]\^4 + 13\/14\ \[ImaginaryI]\ rr\ \[Alpha]\^3 + \(134\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 480\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] - \(960\ \[Alpha]\^2\)\/7 - \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha]\^3 + 19200\)} }], ")"}]}], TraditionalForm]], "Input", CellLabel->"In[27]:=", CellMargins->{{Automatic, Automatic}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { \(\[Alpha]\^4 + 1\/2\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha]\^3 - \(1152\ \[Alpha]\^2\)\/7 + \(10\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 576\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] + 271872\/7\), \(\(-\(1\/4\)\)\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \(912\ \[Alpha]\^2\)\/7 + \(12\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 456\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] + 238080\/7\), \(\(-\(\(416\ \[Alpha]\^2\)\/7\)\) + \(4\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 208\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] + 138048\/7\)}, { \(\(-\(1\/2\)\)\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \(2496\ \[Alpha]\^2\)\/7 + \(48\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 1248\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] + 460800\/7\), \(\[Alpha]\^4 + 1\/2\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha]\^3 - \(1920\ \[Alpha]\^2\)\/7 + \(52\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 960\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] + 405120\/7\), \(\(-\(1\/4\)\)\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \(864\ \[Alpha]\^2\)\/7 + \(8\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 432\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] + 238080\/7\)}, { \(4\/7\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \(3072\ \[Alpha]\^2\)\/7 + \(48\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 1536\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] + 34560\), \(2\/7\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \(2544\ \[Alpha]\^2\)\/7 + \(52\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 1272\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] + 30720\), \(\[Alpha]\^4 + 13\/14\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha]\^3 - \(960\ \[Alpha]\^2\)\/7 + \(134\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 480\/7\ \[ImaginaryI]\ rr\ \[Lambda]\ \[Alpha] + 19200\)} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[27]="] }, Open ]], Cell[TextData["We can see this better with some numbers substituted"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["AB/.{\[Alpha]->1,rr->100}"], "Input", CellLabel->"In[28]:=", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { \(\(56900\ \[ImaginaryI]\ \[Lambda]\)\/7 + \((270727\/7 + \(1350\ \[ImaginaryI]\)\/7)\)\), \(\(45600\ \[ImaginaryI]\ \[Lambda]\)\/7 + \((237168\/7 + \(1025\ \[ImaginaryI]\)\/7)\)\), \(\(20800\ \[ImaginaryI]\ \[Lambda]\)\/7 + \((137632\/7 + \(400\ \[ImaginaryI]\)\/7)\)\)}, { \(\(124800\ \[ImaginaryI]\ \[Lambda]\)\/7 + \((65472 + \(4450\ \[ImaginaryI]\)\/7)\)\), \(\(95300\ \[ImaginaryI]\ \[Lambda]\)\/7 + \((57601 + \(5550\ \[ImaginaryI]\)\/7)\)\), \(\(43200\ \[ImaginaryI]\ \[Lambda]\)\/7 + \((33888 + \(625\ \[ImaginaryI]\)\/7)\)\)}, { \(\(153600\ \[ImaginaryI]\ \[Lambda]\)\/7 + \((238848\/7 + \(5200\ \[ImaginaryI]\)\/7)\)\), \(\(127200\ \[ImaginaryI]\ \[Lambda]\)\/7 + \((212496\/7 + \(5400\ \[ImaginaryI]\)\/7)\)\), \(\(47300\ \[ImaginaryI]\ \[Lambda]\)\/7 + \((133447\/7 + \(14050\ \[ImaginaryI]\)\/7)\)\)} }], ")"}], TraditionalForm]], "Output", CellLabel->"Out[28]="] }, Open ]], Cell[TextData[ "\[Lambda] is apparently an eigen value because we have AB.y = 0, but the \ form is not\nAy = \[Lambda] y . The term \[Lambda] is not just on the \ diagonals. It is really AAy = \[Lambda] BB y. This is the form of a \ generalized eigenvalue problem. Let's see how to construct the problem in \ this form. First get the AA matrix,"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["AA=AB/.\[Lambda]->0"], "Input", CellLabel->"In[29]:=", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { \(\[Alpha]\^4 + 1\/2\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \(1152\ \[Alpha]\^2\)\/7 + \(10\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 271872\/7\), \(\(-\(1\/4\)\)\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \(912\ \[Alpha]\^2\)\/7 + \(12\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 238080\/7\), \(\(-\(\(416\ \[Alpha]\^2\)\/7\)\) + \(4\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 138048\/7\)}, { \(\(-\(1\/2\)\)\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \(2496\ \[Alpha]\^2\)\/7 + \(48\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 460800\/7\), \(\[Alpha]\^4 + 1\/2\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \(1920\ \[Alpha]\^2\)\/7 + \(52\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 405120\/7\), \(\(-\(1\/4\)\)\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - \(864\ \[Alpha]\^2\)\/7 + \(8\ \[ImaginaryI]\ rr\ \[Alpha]\)\/7 + 238080\/7\)}, { \(4\/7\ \[ImaginaryI]\ rr\ \[Alpha]\^3 - 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