Linear Algebra; Stochastic matrix and Steady State vectors

1. Nov 18, 2008

MustangGt94

1. The problem statement, all variables and given/known data

Question: 18. Show that every 2 x 2 stochastic matrix has at least one steady-state vector. Any such matrix can be written as

P = |1-a b |
| a 1-b |

where a and b are constants between 0 and 1. (There are two linearly independent steady-state vectors if a = b = 0. Otherwise, there is only one.)

3. The attempt at a solution

I am guessing that they want me to show that the above matrix has a solution to the equation Px = x so that the steady state vector exists since there is a solution. My solution was that since a and b are other constants between 0 and 1 the matrix would have a solution?

Thank you for the help!

2. Nov 19, 2008

Unco

Hi Mustang,

Consider that if Px = x, then Px - x = 0, i.e., (P - I)x = 0, where I is the 2x2 identity matrix. As a steady state vector (just a certain type of "eigenvector", if you are familiar with the term) is necessarily nonzero, recall how one can use the determinant to determine where the system Ax=0 has a nonzero solution.

Last edited: Nov 19, 2008
3. Nov 19, 2008

MustangGt94

Ah!!! I see what you mean, thanks a lot Unco, much appreciated!