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Cell[CellGroupData[{
Cell["Similarity transforms", "Title",
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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",
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  "\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:  10/98\nMore recent versions of this \
notebook should be available at the web site:\n",
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Suppose that we have our favorite matrix aa.  It is real and will \
have distinct real eigen values.  We see that from the linear ODE problem, it \
would be nice to transform this problem into a simpler problem that has only \
diagonal elements.  Can we do this?  What values will be on the diagonal of \
the simpler matrix?\
\>", "Text"],

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We should verify that these are linearly independent and check to \
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First, they are linearly independent.\
\>", "Text"],

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Now, what will happen if we do not have distinct eigenvalues.?\
\>",
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Recall our matrix that we used to calculate the generalized \
eigenvectors\
\>", "Text"],

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We see that the new matrix is much simpler that the original aa, \
but not exactly diagonal.  This is the expected result for a matrix with \
repeated eigenvalues  However, there is a caveat.

Recall this matrix \
\>", "Text"]
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