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Cell[CellGroupData[{
Cell["Markov Chains", "Section"],

Cell["\<\
The matrix manipulation for the Markov Chain in #1--the ant moving \
on a wire frame.  This MC has a unique steady state distribution.\
\>", "Text"],

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Cell["\<\
Looking at large powers of T gives us the approximate steady state \
distribution.  You can figure it out directly by solving the fixed point \
equation.  (Note on the use of MatrixForm below:  if you want to do anything \
with the output, don't use MatrixForm.  For instance, in the definition of T \
above, it would be a big mistake to use it there...)\
\>", "Text"],

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Cell["\<\
Now the matrix manipulation for the Markov chain in #2, the rat in \
a maze.  Here you have two absorbing states, so we won't get a unique steady \
state distribution.\
\>", "Text"],

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Cell["\<\
Notice that this matrix will never have all positive entries (of \
course this doesn't constitute a proof!).   A real proof would be to argue \
that it is never possible to move from room 1 to any other room, thus the \
first column will always be the column vector {1, 0, 0, 0, 0, 0}, etc.\
\>", \
"Text"],

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Cell["\<\
To figure out the absorbtion probabilities, and consider the \
average number of steps until absorbtion, we need to rewrite the transition \
matrix by putting the absorbtion states in the lower right 'block' of the \
matrix.   Here, the columns (and rows) are numbered 2, 3, 4, 5, 1, 6 for the \
rooms.\
\>", "Text"],

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Cell["\<\
Then the matrix B (see page 126--127) and A become the following:\
\
\>", "Text"],

Cell[CellGroupData[{

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Cell[TextData[{
  "F is called the fundamental matrix.  Here are two ways to interpret the \
entries.  The entry ",
  Cell[BoxData[
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  "represents the average number of times the rat which starts in room 4 is \
in room 4.  The sum of column 4 represents the number of rooms the rat visits \
before hitting room 1 or 6 given that it starts in room 5."
}], "Text"],

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  "This is the absorbtion probability matrix.  For instance, ",
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  " represents the probability that the rat will eventually be absorbed in \
room 1 given it started in room 3.\n\nNotice the wonderful symmetry in all \
these matrices!"
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