# Linear algebra question

1. Feb 24, 2013

### gamerninja213

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

We are given a matrix with the following data

Democrats Republicans Independents Libertarians
Democrats 0.81 0.08 0.16 0.10
Republicans 0.09 0.84 0.05 0.08
Independent 0.06 0.04 0.74 0.04
Libertarians 0.04 0.04 0.05 0.78

2. Relevant equations
Let xn be the vector (Dn, Rn, In, Ln)T. It represents the percentage of representatives of each party after n presidential elections and we shall call it the party distribution

in general xn = (Pn)(x0)

Why is it that for for a significantly large n, say n approaches infinity, no matter what your initial distribution of the electorate is, it does not seem to affect the distribution in the long term.

3. The attempt at a solution

Say, you can choose any vector in R4 for x0 so long as all the entries sum up to 100 and multiplying this by

Pinf =

0.3548 0.3548 0.3548 0.3548
0.3286 0.3286 0.3286 0.3286
0.1570 0.1570 0.1570 0.1570
0.1600 0.1600 0.1600 0.1600

will always give you the following vector:

35.4809
32.8634
15.7046
15.9955

Why is this?

I notice this is true for any Matrix P whose first row has entries that are all the same value second row has entries that are all the same value etc.