[.1 .7 .6 .4]

[.3 .5 .2 .8]

[.4 .9 .9 .5]

If you are currently in state 2 (i.e. row 2), then the probability you will transition to state 4 in the next observation is .8 (i.e. column 4)

Rows are states; columns are probability of being in each state.

Reducing Matrices Video Notes

1) When trying to get a 1 in the first row and first column, multiply the row by the reciprocal of the number you want to be zero. ALL OTHER ROWS REMAIN THE SAME. You may also switch the top row with any other row that has a 1 in the first column… usually doesn’t happen though.

2) Make everything below the 1 zeros. Manipulate the first row so that when you add it to the second row, the first column of row 2 becomes zero. Do the same for row three (i.e. manipulate row 1 again)

3) Find the second leading zero by multiplying the row by the reciprocal of the number you want to be 1 (i.e. the middle entry of the second row). ALL OTHER ROWS REMAIN THE SAME.

4) Manipulate the row in which you created a leading zero and add it to the other row you want to be a zero

5) Repeat

5.3 Video Notes

When there is no leading zero in a column of a completely reduced matrix, then the variable that corresponds with that column is considered arbitrary.

Ex:

[1 -1 0 | -6]

[ 0 0 0 | 2]

Where column 1, 2 and 3 represent variables x, y, and z respectively…

Column 2 does not have a leading zero so variable y is considered arbitrary.

Consistent System = a system that has at least one solution

Inconsistent System = a system that has no solutions

If at any step of the matrix reduction process there is 0’s on the left-hand side and a non-zero number on the right-hand side, then the original system has no solution and is inconsistent

6.2 Video Notes

Inverse of a Matrix Trick (only for 2x2 matrices)

In general, if

A = [ a b ] [ c d ]

where a, b, c, and d represent numbers, then… [ d -b ] A-1 = (1 / ad-bc) x [ -c a ]

…so you multiply (1 / ad-bc) by a matrix where the top left and bottom right entries are exchanged and the top right and bottom left entries are turned negative

8.3 Video Notes

Some matrices start to “stabilize” when you start raising them to higher and higher powers. Instead of P(2), which also can be seen as P squared, we are looking at P(k). At some “k”, every entry in certain matrices become the same.