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Cell[CellGroupData[{
Cell["Diagonizability of Matrices and Jordan Canonical Form", "Title"],

Cell[CellGroupData[{

Cell["Diagonalizable Matrices - Development of Similar Matrices", "Section"],

Cell["\<\
First define your transformation matrix and find its eigenvalues and \
eigenvectors.\
\>", "Text"],

Cell[BoxData[{
    \(Clear[a, b, p, eig, evectors]\), "\n", 
    \(\(a = {{0, 1, \(-1\)}, {4, \(-2\), 0}, {1, 0, \(-2\)}};\) // 
      N\), "\n", 
    \(MatrixForm[a]\), "\n", 
    \(eig = Eigensystem[a]\ \)}], "Input"],

Cell["\<\
Be sure you know how to read the Eigensystem output!!!
Now define the matrix of eigenvectors (p) and verify the form for the similar \
matrix.\
\>", "Text"],

Cell[BoxData[{
    \(Print["\<eigenvector row matrix is \>", 
      MatrixForm[evectors = eig[\([2]\)]]]\), "\n", 
    \(Print["\<eigenvector column matrix (p-matrix) is \>", 
      MatrixForm[p = Transpose[evectors]]]\), "\n", 
    \(Print["\<p inverse is \>", MatrixForm[pinv = Inverse[p]]\ ]\), "\n", 
    \(Print[\*"\"\<A matrix similar to the original matrix is \
\!\(p\^\(-1\)\).a. p = \>\"", MatrixForm[b = pinv . a . p]]\)}], "Input"],

Cell["\<\
Look at the entries in this last matrix compared to the eigenvalues of the \
matrix.\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Non-Diagonizable Matrices", "Section"],

Cell[CellGroupData[{

Cell["Example", "Subsection"],

Cell["Now consider a matrix with defective eigenvalues.", "Text"],

Cell[BoxData[{
    \(Clear[a, p]\), "\[IndentingNewLine]", 
    \(\(a = {{3, 6, \(-7\), 7}, {\(-1\), \(-3\), 1, 0}, {26, 27, \(-31\), 
            29}, {26, 26, \(-27\), 24}};\)\), "\[IndentingNewLine]", 
    \(Print["\<The matrix a = \>", MatrixForm[a]]\), "\[IndentingNewLine]", 
    \(Print["\<The eigensystem for the matrix is \>", 
      eig = Eigensystem[a]]\), "\[IndentingNewLine]", 
    \(\(p = 
        Transpose[
          Table[eig[\([2, i]\)], {i, 1, 
              Length[eig[\([2]\)]]}]];\)\), "\[IndentingNewLine]", 
    \(Print["\<The matrix whose columns are the eigenvectors is \>", 
      MatrixForm[
        p], "\<Does it have an inverse?\>"]\), "\[IndentingNewLine]", 
    \(Inverse[p]\)}], "Input"],

Cell[TextData[{
  "Notice that the matrix p has two zero columns and hence does not possess \
an inverse. In these cases, the next best thing we can do is to reduce it to \
a nearly diagonal matrix using the command ",
  StyleBox["JordanDecomposition",
    FontWeight->"Bold"],
  ". "
}], "Text"],

Cell[BoxData[{
    \(\({p1, b} = JordanDecomposition[a];\)\), "\[IndentingNewLine]", 
    \(Print["\<The matrix p1 is \>", 
      MatrixForm[p1]]\), "\[IndentingNewLine]", 
    \(Print["\<The matrix b is \>", MatrixForm[b]]\), "\[IndentingNewLine]", 
    \(Print[\*"\"\<Does \!\(p1\^\(-1\)\).a.p1 yield b? \>\"", 
      Inverse[p1] . a . p1 \[Equal] b]\)}], "Input"],

Cell["\<\
Notice that two of the columns in the matrix p1 are the two nonzero \
eigenvectors. \
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Exercise ", "Subsection"],

Cell["\<\
To see what the other vectors represent in the example above,consider the \
following problem\
\>", "Text"],

Cell[BoxData[{
    \(\(m = {{3, \(-1\)}, {1, 1}};\)\), "\[IndentingNewLine]", 
    \(Print["\<The differential equation is \>", 
      MatrixForm[{\(x'\)[t], \(y'\)[t]}], "\< = \>", 
      MatrixForm[m], "\<.\>" 
        MatrixForm[{x[t], y[t]}]]\), "\[IndentingNewLine]", 
    \(Print["\<Its eigenvalues and corresponding eigenvectors are \>", 
      Eigensystem[m]]\)}], "Input"],

Cell["\<\
Since we have a defective eigenvalue, we need to find a second solution of \
the following form.\
\>", "Text"],

Cell[BoxData[
    \(sol = 
      Solve[\((m - 2  IdentityMatrix[2])\) . {w1, w2} \[Equal] {1, 1}, {w1, 
          w2}]\)], "Input"],

Cell["If we let w2=0, we get the vector", "Text"],

Cell[BoxData[
    \(MatrixForm[\({w1, w2} /. sol\) /. w2 \[Rule] 0]\)], "Input"],

Cell["\<\
Let's see if this is in any way related to the other column of the p1 matrix \
when we find a Jordan Cannonical form\
\>", "Text"],

Cell[BoxData[{
    \(\({p1, b} = JordanDecomposition[m];\)\), "\[IndentingNewLine]", 
    \(Print["\<The matrix p1 is \>", 
      MatrixForm[p1]]\), "\[IndentingNewLine]", 
    \(Print["\<The matrix b is \>", MatrixForm[b]]\)}], "Input"],

Cell["\<\
Notice that the other column vector of p1 is the same as our {w1,w2}. 
The solution can be written as follows.\
\>", "Text"],

Cell[BoxData[{
    \(\(sone = {1, 1} \[ExponentialE]\^\(2  t\);\)\), "\[IndentingNewLine]", 
    \(\(stwo = {1, 1}\ t\ \[ExponentialE]\^\(2  t\) + {1, 
              0} \[ExponentialE]\^\(2  t\);\)\), "\[IndentingNewLine]", 
    \(generalsolution = k1\ sone + k2\ stwo // Simplify\)}], "Input"],

Cell[TextData[{
  "Does this agree with the solution found with ",
  StyleBox["DSolve",
    FontWeight->"Bold"],
  "? \n",
  StyleBox["NOTE:  ",
    FontWeight->"Bold"],
  "Recall that we were having problem with the DSolve command when I",
  " entered the differential equation in matrix form. I found out that we \
need to use the command ",
  StyleBox["Thread ",
    FontWeight->"Bold"],
  "in Version 4.1 so that ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " will correctly decouple the system of equations."
}], "Text"],

Cell[BoxData[
    \(DSolve[
      Thread[{\(x'\)[t], \(y'\)[t]} \[Equal] m . {x[t], y[t]}], {x[t], y[t]}, 
      t]\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["System of 4 Equations", "Subsection"],

Cell["\<\
Suppose that you have a system of 4 linear differential equations, with one \
repeated eigenvalue\
\>", "Text"],

Cell[BoxData[{
    \(\(m = {{2, 4, \(-6\), 0}, {4, 6, \(-3\), \(-4\)}, {0, 0, 4, 0}, {0, 
            4, \(-6\), 2}};\)\), "\[IndentingNewLine]", 
    \(Print["\<The system is \>", 
      MatrixForm[{\(x1'\)[t], \(x2'\)[t], \(x3'\)[t], \(x4'\)[
            t]}], "\< = \>", 
      MatrixForm[m], "\<.\>" 
        MatrixForm[{x1[t], x2[t], x3[t], x4[t]}]]\), "\[IndentingNewLine]", 
    \(Print["\<The eigenvalues and eigenvectors of m are \>", 
      eig = Eigensystem[m]]\)}], "Input"],

Cell["From this, I can write three solutions.", "Text"],

Cell[BoxData[{
    \(sone = 
      eig[\([2, 
            1]\)] \[ExponentialE]\^\(eig[\([1, 1]\)]\ t\)\), "\
\[IndentingNewLine]", 
    \(stwo = 
      eig[\([2, 
            3]\)] \[ExponentialE]\^\(eig[\([1, 3]\)]\ t\)\), "\
\[IndentingNewLine]", 
    \(sthree = 
      eig[\([2, 4]\)] \[ExponentialE]\^\(eig[\([1, 4]\)]\ t\)\)}], "Input"],

Cell["\<\
Now let's use the JordanDecomposition command to find the vector \
{w1,w2,w3,w4}.\
\>", "Text"],

Cell[BoxData[{
    \(\({p1, b} = JordanDecomposition[m];\)\), "\[IndentingNewLine]", 
    \(MatrixForm /@ {p1, b}\)}], "Input",
  CellTags->"JordanDecomposition"],

Cell["\<\
Contrast this to what we would get if we had used the homework method in the \
previous exercise.\
\>", "Text"],

Cell[BoxData[
    \(sol = 
      Solve[\((m - 2  IdentityMatrix[4])\) . {w1, w2, w3, w4} \[Equal] {1, 0, 
            0, 1}, {w1, w2, w3, w4}]\)], "Input"],

Cell["If we let w4=0, notice the vector we get.", "Text"],

Cell[BoxData[
    \(\({w1, w2, w3, w4} /. sol\) /. w4 \[Rule] 0\)], "Input"],

Cell["\<\
this agrees with the extra vector (noneigenvector) in p1 above.
Now I can write my fourth solution part and my general solution as follows.\
\>", "ExampleText",
  CellTags->"JordanDecomposition"],

Cell[BoxData[{
    \(sfour = \ {\(-1\)/4, 1/4, 0, 0}\ \[ExponentialE]\^\(2  t\) + {1, 0, 0, 
            1}\ t\ \[ExponentialE]\^\(2  t\)\), "\[IndentingNewLine]", 
    \(generalsolution = 
      k1\ sone + k2\ stwo + k3\ sthree + k4\ sfour // Simplify\)}], "Input"],

Cell["How does this compare with the solution using DSolve?", "Text"]
}, Closed]]
}, Closed]]
}, Open  ]]
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End of Mathematica Notebook file.
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