How to set up a Transition Matrix:

Problem: Set up a transition matrix for a row of 8 trees with a 20% transition probability to each neighbor. (Note: no bugs die or get lost.)

About the transition matrix:

· A transition matrix is like a coordinate system. To avoid confusion, it may help to list T1, T2, …T8 across the top and down the sides of the 8x8 matrix. This will allow you to determine how many bugs go to each tree faster.

· There are many patterns inside of a transition matrix. Noticing them can make creating the matrix go faster.

Transition matrix for the problem above:

· Notice how other than the 1^st and 8^th columns, the .2, .6, .2 vertical pattern just shifts down one line each time.

· This pattern also occurs horizontally. (Except for rows 1 and 8.)

The pattern looks like this: .2 .6 .2

.2 .6 .2

.2 .6 .2

etc.

To find the totals after a transition, you must multiply the 8x8 transition matrix by an 8x1 matrix with the current tree populations in it. See below.