12.2. SOME APPLICATIONS OF EIGENVALUES AND EIGENVECTORS 271

Example 12.2.6 Consider the migration matrix

 .6 0 .1.2 .8 0.2 .2 .9

 for locations 1,2, and

3. Suppose initially there are 100 residents in location 1, 200 in location 2 and 400 inlocation 4. Find the population in the three locations after a long time.

You just need to find the eigenvector which goes with the eigenvalue 1 and then nor-malize it so the sum of its entries equals the sum of the entries of the initial vector. Thusyou need to find a solution to

 1 0 00 1 00 0 1

− .6 0 .1

.2 .8 0

.2 .2 .9

 x

yz

=

 000

The augmented matrix is  .4 0 −.1 | 0

−.2 .2 0 | 0−.2 −.2 .1 | 0

and its row reduced echelon form is 1 0 −.25 0

0 1 −.25 00 0 0 0

Therefore, the eigenvectors are

s(

14

14 1

)T

and all that remains is to choose the value of s such that14

s+14

s+ s = 100+200+400

This yields s = 14003 and so the long time limit would equal

14003

 (1/4)(1/4)

1

=

 116.6666666666667116.6666666666667466.6666666666667

 .

You would of course need to round these numbers off. You see that you are not far off afterjust 10 units of time. Therefore, you might consider this as a useful procedure because it isprobably easier to solve a simple system of equations than it is to raise a matrix to a largepower.

Example 12.2.7 Suppose a migration matrix is



15

12

15

14

14

12

1120

14

310

 . Find the comparison

between the populations in the three locations after a long time.