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Notebook[{
Cell["Use of eigenvectors to expand vectors and solve equations", "Title"],

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:  11/98\nMore recent versions of this \
notebook should be available at the web site:\n",
  Cell[BoxData[
      FormBox[
        ButtonBox[\(\(\(http\)\(:\)\) // \(www . nd . edu/\(\(~\)\(mjm\)\)\)/
              vector . expand . nb\),
          ButtonData:>{
            URL[ "http://www.nd.edu/~mjm/vector.expand.nb"], None},
          ButtonStyle->"Hyperlink"], TraditionalForm]]]
}], "Text"],

Cell[CellGroupData[{

Cell["\<\
Expansion of an arbitry vector using the eigenvectors of a matrix\
\
\>", "Subtitle"],

Cell[CellGroupData[{

Cell["First we choose a matrix.  This one is self adjoint.", "Text"],

Cell[BoxData[
    \(aa = \ {{1, \(-1\), 3}, {\(-1\), 0, \(-2\)}, {3, \(-2\), 4}}\)], "Input",\

  CellLabel->"In[228]:=",
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Cell[BoxData[
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  CellLabel->"In[229]:="],

Cell[BoxData[
    \(TraditionalForm\`4\)], "Output",
  CellLabel->"Out[229]="]
}, Open  ]],

Cell["\<\
Here is our arbitrary vector.  We wish to expand it in terms of the \
eigenvalues\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(z = \ {\(-4\), 3, 2}\)], "Input",
  CellLabel->"In[230]:=",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`{\(-4\), 3, 2}\)], "Output",
  CellLabel->"Out[230]="]
}, Open  ]],

Cell["First get the eigenvalues", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(evals = Eigenvalues[aa]\)], "Input",
  CellLabel->"In[231]:=",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`{\(-1\), 3 - \@13, 3 + \@13}\)], "Output",
  CellLabel->"Out[231]="],

Cell["Then get the corresponding eigenvectors", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(evecs = Eigenvectors[aa]\)], "Input",
  CellLabel->"In[232]:=",
  AspectRatioFixed->True],

Cell[BoxData[
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            {\(\(2\ \((3 + \@13)\)\)\/\(11 + 3\ \@13\)\), 
              \(-\(\(5 + \@13\)\/\(11 + 3\ \@13\)\)\), "1"}
            }], ")"}], TraditionalForm]], "Output",
  CellLabel->"Out[232]="]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Now solve the adjoint system for the eigenvalues (not for this \
problem)\
\>", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(evalstrans = Eigenvalues[Transpose[aa]]\)], "Input",
  CellLabel->"In[233]:=",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`{\(-1\), 3 - \@13, 3 + \@13}\)], "Output",
  CellLabel->"Out[233]="],

Cell["and the eigenvectors", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(evecstrans = Eigenvectors[Transpose[aa]]\)], "Input",
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  AspectRatioFixed->True],

Cell[BoxData[
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            }], ")"}], TraditionalForm]], "Output",
  CellLabel->"Out[234]="]
}, Open  ]]
}, Open  ]]
}, Closed]],

Cell["Here is how we write the expansion for the z vector,", "Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(Table[\[Alpha][i], {i, 1, 3}] . {xvec[1], xvec[2], xvec[3]}\)], "Input",\

  CellLabel->"In[235]:="],

Cell[BoxData[
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    \`\(xvec(1)\)\ \(\[Alpha](1)\) + \(xvec(2)\)\ \(\[Alpha](2)\) + 
      \(xvec(3)\)\ \(\[Alpha](3)\)\)], "Output",
  CellLabel->"Out[235]="]
}, Open  ]],

Cell["Now calculate this", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(expand1 = Table[\[Alpha][i], {i, 1, 3}] . evecs\)], "Input",
  CellLabel->"In[236]:="],

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        \(2\ \((\(-3\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 3\ \@13\) + 
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        \(\((\(-5\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 3\ \@13\) - 
        \(\((5 + \@13)\)\ \(\[Alpha](3)\)\)\/\(11 + 3\ \@13\), 
      \[Alpha](1) + \[Alpha](2) + \[Alpha](3)}\)], "Output",
  CellLabel->"Out[236]="]
}, Open  ]],

Cell[TextData[
"Now we have the equation, z =Sum[\[Alpha][i] x[i]] for the expansion.  We \
evaluate the \[Alpha][i]'s by forming the inner product of both sides of the \
equation with the adjoint eigenvectors (which are the same as the regular \
eigenvectors).  \n\nHere is the right side"], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[237]:="],

Cell[BoxData[
    \(TraditionalForm
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        \(2\ \((\(-3\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 3\ \@13\) - 
        \(\((\(-5\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 3\ \@13\) + 
        \[Alpha](2) - \(\((5 + \@13)\)\ \(\[Alpha](3)\)\)\/\(11 + 3\ \@13\) - 
        \(2\ \((3 + \@13)\)\ \(\[Alpha](3)\)\)\/\(11 + 3\ \@13\) + 
        \[Alpha](3), 
      \[Alpha](1) + \[Alpha](2) + \[Alpha](3) + 
        \(2\ \((\(-3\) + \@13)\)\ 
            \((\(-\(\[Alpha](1)\)\) + 
                \(2\ \((\(-3\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 
                    3\ \@13\) + 
                \(2\ \((3 + \@13)\)\ \(\[Alpha](3)\)\)\/\(11 + 3\ \@13\))
              \)\)\/\(\(-11\) + 3\ \@13\) - 
        \(\((\(-5\) + \@13)\)\ 
            \((\[Alpha](1) - 
                \(\((\(-5\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 
                    3\ \@13\) - 
                \(\((5 + \@13)\)\ \(\[Alpha](3)\)\)\/\(11 + 3\ \@13\))
              \)\)\/\(\(-11\) + 3\ \@13\), 
      \[Alpha](1) + \[Alpha](2) + \[Alpha](3) + 
        \(2\ \((3 + \@13)\)\ 
            \((\(-\(\[Alpha](1)\)\) + 
                \(2\ \((\(-3\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 
                    3\ \@13\) + 
                \(2\ \((3 + \@13)\)\ \(\[Alpha](3)\)\)\/\(11 + 3\ \@13\))
              \)\)\/\(11 + 3\ \@13\) - 
        \(\((5 + \@13)\)\ 
            \((\[Alpha](1) - 
                \(\((\(-5\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 
                    3\ \@13\) - 
                \(\((5 + \@13)\)\ \(\[Alpha](3)\)\)\/\(11 + 3\ \@13\))
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  CellLabel->"Out[237]="]
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Cell[CellGroupData[{

Cell[BoxData[
    \(expand3 = Simplify[expand2]\)], "Input",
  CellLabel->"In[238]:="],

Cell[BoxData[
    \(TraditionalForm
    \`{3\ \(\[Alpha](1)\), 
      \(2\ \((\(-13\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 3\ \@13\), 
      \(2\ \((13 + \@13)\)\ \(\[Alpha](3)\)\)\/\(11 + 3\ \@13\)}\)], "Output",\

  CellLabel->"Out[238]="]
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Cell["Here is the left side", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(left1 = evecs . z\)], "Input",
  CellLabel->"In[239]:="],

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      2 - \(8\ \((3 + \@13)\)\)\/\(11 + 3\ \@13\) - 
        \(3\ \((5 + \@13)\)\)\/\(11 + 3\ \@13\)}\)], "Output",
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Cell[TextData[
"Now get the \[Alpha][i]'s by equating the two sides and solving"], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[240]:="],

Cell[BoxData[
    \(TraditionalForm
    \`{9, 2 - \(3\ \((\(-5\) + \@13)\)\)\/\(\(-11\) + 3\ \@13\) - 
          \(8\ \((\(-3\) + \@13)\)\)\/\(\(-11\) + 3\ \@13\), 
        2 - \(8\ \((3 + \@13)\)\)\/\(11 + 3\ \@13\) - 
          \(3\ \((5 + \@13)\)\)\/\(11 + 3\ \@13\)} == {3\ \(\[Alpha](1)\), 
        \(2\ \((\(-13\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 3\ \@13\), 
        \(2\ \((13 + \@13)\)\ \(\[Alpha](3)\)\)\/\(11 + 3\ \@13\)}\)], 
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  CellLabel->"Out[240]="]
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Cell[TextData["Note that each equation contains only 1 \[Alpha][i]."], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    \(TraditionalForm
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        \[Alpha](2) \[Rule] 
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        \[Alpha](3) \[Rule] 
          \(-\(\(17 + 5\ \@13\)\/\(2\ \((13 + \@13)\)\)\)\)}}\)], "Output",
  CellLabel->"Out[241]="]
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Cell["Now let's check the result", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(test1 = Table[\[Alpha][i], {i, 1, 3}] . evecs\)], "Input",
  CellLabel->"In[242]:="],

Cell[BoxData[
    \(TraditionalForm
    \`{\(-\(\[Alpha](1)\)\) + 
        \(2\ \((\(-3\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 3\ \@13\) + 
        \(2\ \((3 + \@13)\)\ \(\[Alpha](3)\)\)\/\(11 + 3\ \@13\), 
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        \(\((\(-5\) + \@13)\)\ \(\[Alpha](2)\)\)\/\(\(-11\) + 3\ \@13\) - 
        \(\((5 + \@13)\)\ \(\[Alpha](3)\)\)\/\(11 + 3\ \@13\), 
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  CellLabel->"Out[242]="]
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Cell["Now substitute", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(test2 = test1 /. ans[\([1]\)]\)], "Input",
  CellLabel->"In[243]:="],

Cell[BoxData[
    \(TraditionalForm
    \`{\(-3\) - 
        \(\((\(-3\) + \@13)\)\ 
            \((\(-17\) + 5\ \@13)\)\)\/\(\((\(-13\) + \@13)\)\ 
            \((\(-11\) + 3\ \@13)\)\) - 
        \(\((3 + \@13)\)\ \((17 + 5\ \@13)\)\)\/\(\((13 + \@13)\)\ 
            \((11 + 3\ \@13)\)\), 
      3 + \(\((\(-5\) + \@13)\)\ \((\(-17\) + 5\ \@13)\)\)\/\(2\ 
            \((\(-13\) + \@13)\)\ \((\(-11\) + 3\ \@13)\)\) + 
        \(\((5 + \@13)\)\ \((17 + 5\ \@13)\)\)\/\(2\ \((13 + \@13)\)\ 
            \((11 + 3\ \@13)\)\), 
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Cell[BoxData[
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  CellLabel->"In[244]:="],

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Cell["Recall z,", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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  CellLabel->"In[245]:="],

Cell[BoxData[
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  CellLabel->"Out[245]="]
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Cell["It, of course, matches.", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["We choose another matrix.  This one is not self adjoint.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
Here is our arbitrary vector.  We wish to expand it in terms of the \
eigenvalues\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(z = \ {\(-1\), 3, 5}\)], "Input",
  CellLabel->"In[127]:=",
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["Now solve the adjoint system for the eigenvalues ", "Text",
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Cell[CellGroupData[{

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Cell["Now calculate this", "Text"],

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evaluate the \[Alpha][i]'s by forming the inner product of both sides of the \
equation with the adjoint eigenvectors.\n\nHere is the right side"], "Text"],

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Cell["Here is the left side", "Text"],

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Cell[BoxData[
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Cell[TextData[
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Cell[BoxData[
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Cell[TextData["Note that each equation contains only 1 \[Alpha][i]."], "Text"],

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Cell[BoxData[
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Cell["Now let's check the result", "Text"],

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Cell[CellGroupData[{

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Cell["Recall z,", "Text"],

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Cell[BoxData[
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Cell["It, of course, matches.", "Text"]
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Cell["Solution of a system of equations by eigenvector expansion", "Subtitle",
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Cell[CellGroupData[{

Cell["We choose again a matrix", "Text",
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Cell["\<\
Note that I use the N[ ] command to make every thing more compact \
and faster.  \
\>", "Text",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell["So we are safe as far as getting a solution", "Text",
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Cell["Let's find the solution, voila!!", "Text",
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Cell["\<\
But this is a graduate course so we have to take an circuitous \
journey\
\>", "Text",
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Cell["First get the eigenvalues", "Text",
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